Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, B

Percentage Accurate: 99.8% → 99.8%

Time: 15.1s

Alternatives: 23

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = (((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
}

Python

def code(x, y, z, t, a, b, c, i):
	return (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = (((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
}

Python

def code(x, y, z, t, a, b, c, i):
	return (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = (((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
}

Python

def code(x, y, z, t, a, b, c, i):
	return (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 45.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \left(\left(\left(\left(t\_1 + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ t_3 := t\_1 + y \cdot i\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+292}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 200:\\ \;\;\;\;\mathsf{fma}\left(\log c, b + -0.5, z\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t + \left(a + b \cdot \log c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* x (log y)))
        (t_2 (+ (+ (+ (+ (+ t_1 z) t) a) (* (- b 0.5) (log c))) (* y i)))
        (t_3 (+ t_1 (* y i))))
   (if (<= t_2 -5e+292)
     t_3
     (if (<= t_2 200.0)
       (fma (log c) (+ b -0.5) z)
       (if (<= t_2 INFINITY) (+ t (+ a (* b (log c)))) t_3)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x * log(y);
	double t_2 = ((((t_1 + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double t_3 = t_1 + (y * i);
	double tmp;
	if (t_2 <= -5e+292) {
		tmp = t_3;
	} else if (t_2 <= 200.0) {
		tmp = fma(log(c), (b + -0.5), z);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t + (a + (b * log(c)));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x * log(y))
	t_2 = Float64(Float64(Float64(Float64(Float64(t_1 + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	t_3 = Float64(t_1 + Float64(y * i))
	tmp = 0.0
	if (t_2 <= -5e+292)
		tmp = t_3;
	elseif (t_2 <= 200.0)
		tmp = fma(log(c), Float64(b + -0.5), z);
	elseif (t_2 <= Inf)
		tmp = Float64(t + Float64(a + Float64(b * log(c))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(t$95$1 + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+292], t$95$3, If[LessEqual[t$95$2, 200.0], N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + z), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(t + N[(a + N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -4.9999999999999996e292 or +inf.0 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} + y \cdot i \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \log y} + y \cdot i \]

      2. log-lowering-log.f64 85.6

         \[\leadsto x \cdot \color{blue}{\log y} + y \cdot i \]

   5. Simplified 85.6%

      \[\leadsto \color{blue}{x \cdot \log y} + y \cdot i \]

   if -4.9999999999999996e292 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 200

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\color{blue}{z} + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]
   4. Step-by-step derivation

   5. Simplified 50.6%

      \[\leadsto \left(\color{blue}{z} + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{z + \log c \cdot \left(b - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + z} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z\right)} \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z\right) \]

      6. +-lowering-+.f64 42.1

         \[\leadsto \mathsf{fma}\left(\log c, \color{blue}{b + -0.5}, z\right) \]

   8. Simplified 42.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log c, b + -0.5, z\right)} \]

   if 200 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < +inf.0

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\color{blue}{z \cdot \left(1 + \frac{x \cdot \log y}{z}\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{z \cdot \left(1 + \frac{x \cdot \log y}{z}\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\left(z \cdot \color{blue}{\left(\frac{x \cdot \log y}{z} + 1\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      3. associate-/l* N/A

         \[\leadsto \left(\left(\left(z \cdot \left(\color{blue}{x \cdot \frac{\log y}{z}} + 1\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\left(z \cdot \color{blue}{\mathsf{fma}\left(x, \frac{\log y}{z}, 1\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(z \cdot \mathsf{fma}\left(x, \color{blue}{\frac{\log y}{z}}, 1\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      6. log-lowering-log.f64 90.7

         \[\leadsto \left(\left(\left(z \cdot \mathsf{fma}\left(x, \frac{\color{blue}{\log y}}{z}, 1\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   5. Simplified 90.7%

      \[\leadsto \left(\left(\left(\color{blue}{z \cdot \mathsf{fma}\left(x, \frac{\log y}{z}, 1\right)} + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{t + \left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{t + \left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto t + \color{blue}{\left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      5. +-commutative N/A

         \[\leadsto t + \left(\color{blue}{\left(\left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right) + z\right)} + a\right) \]

      6. associate-+l+ N/A

         \[\leadsto t + \left(\color{blue}{\left(i \cdot y + \left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)\right)} + a\right) \]

      7. *-commutative N/A

         \[\leadsto t + \left(\left(\color{blue}{y \cdot i} + \left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)\right) + a\right) \]

      8. +-commutative N/A

         \[\leadsto t + \left(\left(y \cdot i + \color{blue}{\left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)}\right) + a\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto t + \left(\color{blue}{\mathsf{fma}\left(y, i, z + \log c \cdot \left(b - \frac{1}{2}\right)\right)} + a\right) \]

      10. +-commutative N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + z}\right) + a\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z\right)}\right) + a\right) \]

      12. log-lowering-log.f64 N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z\right)\right) + a\right) \]

      13. sub-neg N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z\right)\right) + a\right) \]

      14. metadata-eval N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z\right)\right) + a\right) \]

      15. +-lowering-+.f64 84.1

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + -0.5}, z\right)\right) + a\right) \]

   8. Simplified 84.1%

      \[\leadsto \color{blue}{t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right) + a\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto t + \left(\color{blue}{b \cdot \log c} + a\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto t + \left(\color{blue}{\log c \cdot b} + a\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto t + \left(\color{blue}{\log c \cdot b} + a\right) \]

       3. log-lowering-log.f64 49.0

          \[\leadsto t + \left(\color{blue}{\log c} \cdot b + a\right) \]

   11. Simplified 49.0%

       \[\leadsto t + \left(\color{blue}{\log c \cdot b} + a\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -5 \cdot 10^{+292}:\\ \;\;\;\;x \cdot \log y + y \cdot i\\ \mathbf{elif}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq 200:\\ \;\;\;\;\mathsf{fma}\left(\log c, b + -0.5, z\right)\\ \mathbf{elif}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq \infty:\\ \;\;\;\;t + \left(a + b \cdot \log c\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y + y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 44.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+303}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \left(y \cdot i\right), \frac{1}{a}, a\right)\\ \mathbf{elif}\;t\_1 \leq 200:\\ \;\;\;\;\mathsf{fma}\left(\log c, b + -0.5, z\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t + \left(a + b \cdot \log c\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 -4e+303)
     (fma (* a (* y i)) (/ 1.0 a) a)
     (if (<= t_1 200.0)
       (fma (log c) (+ b -0.5) z)
       (if (<= t_1 INFINITY) (+ t (+ a (* b (log c)))) (* y i))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -4e+303) {
		tmp = fma((a * (y * i)), (1.0 / a), a);
	} else if (t_1 <= 200.0) {
		tmp = fma(log(c), (b + -0.5), z);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t + (a + (b * log(c)));
	} else {
		tmp = y * i;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= -4e+303)
		tmp = fma(Float64(a * Float64(y * i)), Float64(1.0 / a), a);
	elseif (t_1 <= 200.0)
		tmp = fma(log(c), Float64(b + -0.5), z);
	elseif (t_1 <= Inf)
		tmp = Float64(t + Float64(a + Float64(b * log(c))));
	else
		tmp = Float64(y * i);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+303], N[(N[(a * N[(y * i), $MachinePrecision]), $MachinePrecision] * N[(1.0 / a), $MachinePrecision] + a), $MachinePrecision], If[LessEqual[t$95$1, 200.0], N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + z), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(t + N[(a + N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * i), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -4e303

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)} \]

      2. sub-neg N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{-1}\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right) \cdot \left(-1 \cdot a\right) + -1 \cdot \left(-1 \cdot a\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right)} + -1 \cdot \left(-1 \cdot a\right) \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot a\right) \cdot -1\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}} + -1 \cdot \left(-1 \cdot a\right) \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(-1 \cdot a\right)\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot -1\right) \cdot a\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\color{blue}{1} \cdot a\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      10. *-lft-identity N/A

          \[\leadsto \color{blue}{a} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      11. associate-*r* N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{\left(-1 \cdot -1\right) \cdot a} \]

      12. metadata-eval N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{1} \cdot a \]

      13. *-lft-identity N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{a} \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, t\right)\right)}{a}, a\right)} \]

   6. Taylor expanded in i around inf

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{i \cdot y}{a}}, a\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{i \cdot y}{a}}, a\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{y \cdot i}}{a}, a\right) \]

      3. *-lowering-*.f64 94.5

         \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{y \cdot i}}{a}, a\right) \]

   8. Simplified 94.5%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{y \cdot i}{a}}, a\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{a \cdot \left(y \cdot i\right)}{a}} + a \]

      2. div-inv N/A

         \[\leadsto \color{blue}{\left(a \cdot \left(y \cdot i\right)\right) \cdot \frac{1}{a}} + a \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \left(y \cdot i\right), \frac{1}{a}, a\right)} \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(y \cdot i\right) \cdot a}, \frac{1}{a}, a\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(y \cdot i\right) \cdot a}, \frac{1}{a}, a\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(y \cdot i\right)} \cdot a, \frac{1}{a}, a\right) \]

      7. /-lowering-/.f64 95.1

         \[\leadsto \mathsf{fma}\left(\left(y \cdot i\right) \cdot a, \color{blue}{\frac{1}{a}}, a\right) \]

   10. Applied egg-rr 95.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y \cdot i\right) \cdot a, \frac{1}{a}, a\right)} \]

   if -4e303 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 200

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\color{blue}{z} + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]
   4. Step-by-step derivation

   5. Simplified 48.5%

      \[\leadsto \left(\color{blue}{z} + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{z + \log c \cdot \left(b - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + z} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z\right)} \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z\right) \]

      6. +-lowering-+.f64 39.9

         \[\leadsto \mathsf{fma}\left(\log c, \color{blue}{b + -0.5}, z\right) \]

   8. Simplified 39.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log c, b + -0.5, z\right)} \]

   if 200 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < +inf.0

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\color{blue}{z \cdot \left(1 + \frac{x \cdot \log y}{z}\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{z \cdot \left(1 + \frac{x \cdot \log y}{z}\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\left(z \cdot \color{blue}{\left(\frac{x \cdot \log y}{z} + 1\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      3. associate-/l* N/A

         \[\leadsto \left(\left(\left(z \cdot \left(\color{blue}{x \cdot \frac{\log y}{z}} + 1\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\left(z \cdot \color{blue}{\mathsf{fma}\left(x, \frac{\log y}{z}, 1\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(z \cdot \mathsf{fma}\left(x, \color{blue}{\frac{\log y}{z}}, 1\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      6. log-lowering-log.f64 90.7

         \[\leadsto \left(\left(\left(z \cdot \mathsf{fma}\left(x, \frac{\color{blue}{\log y}}{z}, 1\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   5. Simplified 90.7%

      \[\leadsto \left(\left(\left(\color{blue}{z \cdot \mathsf{fma}\left(x, \frac{\log y}{z}, 1\right)} + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{t + \left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{t + \left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto t + \color{blue}{\left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      5. +-commutative N/A

         \[\leadsto t + \left(\color{blue}{\left(\left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right) + z\right)} + a\right) \]

      6. associate-+l+ N/A

         \[\leadsto t + \left(\color{blue}{\left(i \cdot y + \left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)\right)} + a\right) \]

      7. *-commutative N/A

         \[\leadsto t + \left(\left(\color{blue}{y \cdot i} + \left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)\right) + a\right) \]

      8. +-commutative N/A

         \[\leadsto t + \left(\left(y \cdot i + \color{blue}{\left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)}\right) + a\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto t + \left(\color{blue}{\mathsf{fma}\left(y, i, z + \log c \cdot \left(b - \frac{1}{2}\right)\right)} + a\right) \]

      10. +-commutative N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + z}\right) + a\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z\right)}\right) + a\right) \]

      12. log-lowering-log.f64 N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z\right)\right) + a\right) \]

      13. sub-neg N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z\right)\right) + a\right) \]

      14. metadata-eval N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z\right)\right) + a\right) \]

      15. +-lowering-+.f64 84.1

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + -0.5}, z\right)\right) + a\right) \]

   8. Simplified 84.1%

      \[\leadsto \color{blue}{t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right) + a\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto t + \left(\color{blue}{b \cdot \log c} + a\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto t + \left(\color{blue}{\log c \cdot b} + a\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto t + \left(\color{blue}{\log c \cdot b} + a\right) \]

       3. log-lowering-log.f64 49.0

          \[\leadsto t + \left(\color{blue}{\log c} \cdot b + a\right) \]

   11. Simplified 49.0%

       \[\leadsto t + \left(\color{blue}{\log c \cdot b} + a\right) \]

   if +inf.0 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{i \cdot y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 20.2

         \[\leadsto \color{blue}{i \cdot y} \]

   5. Simplified 20.2%

      \[\leadsto \color{blue}{i \cdot y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -4 \cdot 10^{+303}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \left(y \cdot i\right), \frac{1}{a}, a\right)\\ \mathbf{elif}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq 200:\\ \;\;\;\;\mathsf{fma}\left(\log c, b + -0.5, z\right)\\ \mathbf{elif}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq \infty:\\ \;\;\;\;t + \left(a + b \cdot \log c\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 44.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+303}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \left(y \cdot i\right), \frac{1}{a}, a\right)\\ \mathbf{elif}\;t\_1 \leq 200:\\ \;\;\;\;\mathsf{fma}\left(\log c, b + -0.5, z\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+271}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, a\right)\\ \mathbf{else}:\\ \;\;\;\;a + \mathsf{fma}\left(y, i, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 -4e+303)
     (fma (* a (* y i)) (/ 1.0 a) a)
     (if (<= t_1 200.0)
       (fma (log c) (+ b -0.5) z)
       (if (<= t_1 1e+271) (fma x (log y) a) (+ a (fma y i t)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -4e+303) {
		tmp = fma((a * (y * i)), (1.0 / a), a);
	} else if (t_1 <= 200.0) {
		tmp = fma(log(c), (b + -0.5), z);
	} else if (t_1 <= 1e+271) {
		tmp = fma(x, log(y), a);
	} else {
		tmp = a + fma(y, i, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= -4e+303)
		tmp = fma(Float64(a * Float64(y * i)), Float64(1.0 / a), a);
	elseif (t_1 <= 200.0)
		tmp = fma(log(c), Float64(b + -0.5), z);
	elseif (t_1 <= 1e+271)
		tmp = fma(x, log(y), a);
	else
		tmp = Float64(a + fma(y, i, t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+303], N[(N[(a * N[(y * i), $MachinePrecision]), $MachinePrecision] * N[(1.0 / a), $MachinePrecision] + a), $MachinePrecision], If[LessEqual[t$95$1, 200.0], N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + z), $MachinePrecision], If[LessEqual[t$95$1, 1e+271], N[(x * N[Log[y], $MachinePrecision] + a), $MachinePrecision], N[(a + N[(y * i + t), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -4e303

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)} \]

      2. sub-neg N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{-1}\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right) \cdot \left(-1 \cdot a\right) + -1 \cdot \left(-1 \cdot a\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right)} + -1 \cdot \left(-1 \cdot a\right) \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot a\right) \cdot -1\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}} + -1 \cdot \left(-1 \cdot a\right) \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(-1 \cdot a\right)\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot -1\right) \cdot a\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\color{blue}{1} \cdot a\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      10. *-lft-identity N/A

          \[\leadsto \color{blue}{a} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      11. associate-*r* N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{\left(-1 \cdot -1\right) \cdot a} \]

      12. metadata-eval N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{1} \cdot a \]

      13. *-lft-identity N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{a} \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, t\right)\right)}{a}, a\right)} \]

   6. Taylor expanded in i around inf

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{i \cdot y}{a}}, a\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{i \cdot y}{a}}, a\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{y \cdot i}}{a}, a\right) \]

      3. *-lowering-*.f64 94.5

         \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{y \cdot i}}{a}, a\right) \]

   8. Simplified 94.5%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{y \cdot i}{a}}, a\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{a \cdot \left(y \cdot i\right)}{a}} + a \]

      2. div-inv N/A

         \[\leadsto \color{blue}{\left(a \cdot \left(y \cdot i\right)\right) \cdot \frac{1}{a}} + a \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \left(y \cdot i\right), \frac{1}{a}, a\right)} \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(y \cdot i\right) \cdot a}, \frac{1}{a}, a\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(y \cdot i\right) \cdot a}, \frac{1}{a}, a\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(y \cdot i\right)} \cdot a, \frac{1}{a}, a\right) \]

      7. /-lowering-/.f64 95.1

         \[\leadsto \mathsf{fma}\left(\left(y \cdot i\right) \cdot a, \color{blue}{\frac{1}{a}}, a\right) \]

   10. Applied egg-rr 95.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y \cdot i\right) \cdot a, \frac{1}{a}, a\right)} \]

   if -4e303 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 200

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\color{blue}{z} + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]
   4. Step-by-step derivation

   5. Simplified 48.5%

      \[\leadsto \left(\color{blue}{z} + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{z + \log c \cdot \left(b - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + z} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z\right)} \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z\right) \]

      6. +-lowering-+.f64 39.9

         \[\leadsto \mathsf{fma}\left(\log c, \color{blue}{b + -0.5}, z\right) \]

   8. Simplified 39.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log c, b + -0.5, z\right)} \]

   if 200 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 9.99999999999999953e270

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)} \]

      2. sub-neg N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{-1}\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right) \cdot \left(-1 \cdot a\right) + -1 \cdot \left(-1 \cdot a\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right)} + -1 \cdot \left(-1 \cdot a\right) \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot a\right) \cdot -1\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}} + -1 \cdot \left(-1 \cdot a\right) \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(-1 \cdot a\right)\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot -1\right) \cdot a\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\color{blue}{1} \cdot a\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      10. *-lft-identity N/A

          \[\leadsto \color{blue}{a} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      11. associate-*r* N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{\left(-1 \cdot -1\right) \cdot a} \]

      12. metadata-eval N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{1} \cdot a \]

      13. *-lft-identity N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{a} \]

   5. Simplified 71.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, t\right)\right)}{a}, a\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{x \cdot \log y}{a}}, a\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{x \cdot \log y}{a}}, a\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{x \cdot \log y}}{a}, a\right) \]

      3. log-lowering-log.f64 28.2

         \[\leadsto \mathsf{fma}\left(a, \frac{x \cdot \color{blue}{\log y}}{a}, a\right) \]

   8. Simplified 28.2%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{x \cdot \log y}{a}}, a\right) \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{a + x \cdot \log y} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{x \cdot \log y + a} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, a\right)} \]

       3. log-lowering-log.f64 35.5

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\log y}, a\right) \]

   11. Simplified 35.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, a\right)} \]

   if 9.99999999999999953e270 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)} \]

      2. sub-neg N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{-1}\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right) \cdot \left(-1 \cdot a\right) + -1 \cdot \left(-1 \cdot a\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right)} + -1 \cdot \left(-1 \cdot a\right) \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot a\right) \cdot -1\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}} + -1 \cdot \left(-1 \cdot a\right) \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(-1 \cdot a\right)\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot -1\right) \cdot a\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\color{blue}{1} \cdot a\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      10. *-lft-identity N/A

          \[\leadsto \color{blue}{a} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      11. associate-*r* N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{\left(-1 \cdot -1\right) \cdot a} \]

      12. metadata-eval N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{1} \cdot a \]

      13. *-lft-identity N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{a} \]

   5. Simplified 72.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, t\right)\right)}{a}, a\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \color{blue}{t}}{a}, a\right) \]
   7. Step-by-step derivation

   8. Simplified 57.6%

      \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \color{blue}{t}}{a}, a\right) \]

   9. Taylor expanded in z around 0

      \[\leadsto \color{blue}{a + \left(t + i \cdot y\right)} \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{a + \left(t + i \cdot y\right)} \]

       2. +-commutative N/A

          \[\leadsto a + \color{blue}{\left(i \cdot y + t\right)} \]

       3. *-commutative N/A

          \[\leadsto a + \left(\color{blue}{y \cdot i} + t\right) \]

       4. accelerator-lowering-fma.f64 57.4

          \[\leadsto a + \color{blue}{\mathsf{fma}\left(y, i, t\right)} \]

   11. Simplified 57.4%

       \[\leadsto \color{blue}{a + \mathsf{fma}\left(y, i, t\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -4 \cdot 10^{+303}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \left(y \cdot i\right), \frac{1}{a}, a\right)\\ \mathbf{elif}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq 200:\\ \;\;\;\;\mathsf{fma}\left(\log c, b + -0.5, z\right)\\ \mathbf{elif}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq 10^{+271}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, a\right)\\ \mathbf{else}:\\ \;\;\;\;a + \mathsf{fma}\left(y, i, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 36.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+303}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;t\_1 \leq -20:\\ \;\;\;\;z + t\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t + a\\ \mathbf{else}:\\ \;\;\;\;y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 -4e+303)
     (* y i)
     (if (<= t_1 -20.0) (+ z t) (if (<= t_1 INFINITY) (+ t a) (* y i))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -4e+303) {
		tmp = y * i;
	} else if (t_1 <= -20.0) {
		tmp = z + t;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t + a;
	} else {
		tmp = y * i;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
	double tmp;
	if (t_1 <= -4e+303) {
		tmp = y * i;
	} else if (t_1 <= -20.0) {
		tmp = z + t;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t + a;
	} else {
		tmp = y * i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)
	tmp = 0
	if t_1 <= -4e+303:
		tmp = y * i
	elif t_1 <= -20.0:
		tmp = z + t
	elif t_1 <= math.inf:
		tmp = t + a
	else:
		tmp = y * i
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= -4e+303)
		tmp = Float64(y * i);
	elseif (t_1 <= -20.0)
		tmp = Float64(z + t);
	elseif (t_1 <= Inf)
		tmp = Float64(t + a);
	else
		tmp = Float64(y * i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	tmp = 0.0;
	if (t_1 <= -4e+303)
		tmp = y * i;
	elseif (t_1 <= -20.0)
		tmp = z + t;
	elseif (t_1 <= Inf)
		tmp = t + a;
	else
		tmp = y * i;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+303], N[(y * i), $MachinePrecision], If[LessEqual[t$95$1, -20.0], N[(z + t), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(t + a), $MachinePrecision], N[(y * i), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -4e303 or +inf.0 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{i \cdot y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 94.7

         \[\leadsto \color{blue}{i \cdot y} \]

   5. Simplified 94.7%

      \[\leadsto \color{blue}{i \cdot y} \]

   if -4e303 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -20

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\color{blue}{z \cdot \left(1 + \frac{x \cdot \log y}{z}\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{z \cdot \left(1 + \frac{x \cdot \log y}{z}\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\left(z \cdot \color{blue}{\left(\frac{x \cdot \log y}{z} + 1\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      3. associate-/l* N/A

         \[\leadsto \left(\left(\left(z \cdot \left(\color{blue}{x \cdot \frac{\log y}{z}} + 1\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\left(z \cdot \color{blue}{\mathsf{fma}\left(x, \frac{\log y}{z}, 1\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(z \cdot \mathsf{fma}\left(x, \color{blue}{\frac{\log y}{z}}, 1\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      6. log-lowering-log.f64 86.6

         \[\leadsto \left(\left(\left(z \cdot \mathsf{fma}\left(x, \frac{\color{blue}{\log y}}{z}, 1\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   5. Simplified 86.6%

      \[\leadsto \left(\left(\left(\color{blue}{z \cdot \mathsf{fma}\left(x, \frac{\log y}{z}, 1\right)} + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{t + \left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{t + \left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto t + \color{blue}{\left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      5. +-commutative N/A

         \[\leadsto t + \left(\color{blue}{\left(\left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right) + z\right)} + a\right) \]

      6. associate-+l+ N/A

         \[\leadsto t + \left(\color{blue}{\left(i \cdot y + \left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)\right)} + a\right) \]

      7. *-commutative N/A

         \[\leadsto t + \left(\left(\color{blue}{y \cdot i} + \left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)\right) + a\right) \]

      8. +-commutative N/A

         \[\leadsto t + \left(\left(y \cdot i + \color{blue}{\left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)}\right) + a\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto t + \left(\color{blue}{\mathsf{fma}\left(y, i, z + \log c \cdot \left(b - \frac{1}{2}\right)\right)} + a\right) \]

      10. +-commutative N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + z}\right) + a\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z\right)}\right) + a\right) \]

      12. log-lowering-log.f64 N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z\right)\right) + a\right) \]

      13. sub-neg N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z\right)\right) + a\right) \]

      14. metadata-eval N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z\right)\right) + a\right) \]

      15. +-lowering-+.f64 84.5

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + -0.5}, z\right)\right) + a\right) \]

   8. Simplified 84.5%

      \[\leadsto \color{blue}{t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right) + a\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto t + \color{blue}{z} \]
   10. Step-by-step derivation

   11. Simplified 37.6%

       \[\leadsto t + \color{blue}{z} \]

   if -20 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < +inf.0

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\color{blue}{z \cdot \left(1 + \frac{x \cdot \log y}{z}\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{z \cdot \left(1 + \frac{x \cdot \log y}{z}\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\left(z \cdot \color{blue}{\left(\frac{x \cdot \log y}{z} + 1\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      3. associate-/l* N/A

         \[\leadsto \left(\left(\left(z \cdot \left(\color{blue}{x \cdot \frac{\log y}{z}} + 1\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\left(z \cdot \color{blue}{\mathsf{fma}\left(x, \frac{\log y}{z}, 1\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(z \cdot \mathsf{fma}\left(x, \color{blue}{\frac{\log y}{z}}, 1\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      6. log-lowering-log.f64 90.8

         \[\leadsto \left(\left(\left(z \cdot \mathsf{fma}\left(x, \frac{\color{blue}{\log y}}{z}, 1\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   5. Simplified 90.8%

      \[\leadsto \left(\left(\left(\color{blue}{z \cdot \mathsf{fma}\left(x, \frac{\log y}{z}, 1\right)} + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{t + \left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{t + \left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto t + \color{blue}{\left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      5. +-commutative N/A

         \[\leadsto t + \left(\color{blue}{\left(\left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right) + z\right)} + a\right) \]

      6. associate-+l+ N/A

         \[\leadsto t + \left(\color{blue}{\left(i \cdot y + \left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)\right)} + a\right) \]

      7. *-commutative N/A

         \[\leadsto t + \left(\left(\color{blue}{y \cdot i} + \left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)\right) + a\right) \]

      8. +-commutative N/A

         \[\leadsto t + \left(\left(y \cdot i + \color{blue}{\left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)}\right) + a\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto t + \left(\color{blue}{\mathsf{fma}\left(y, i, z + \log c \cdot \left(b - \frac{1}{2}\right)\right)} + a\right) \]

      10. +-commutative N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + z}\right) + a\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z\right)}\right) + a\right) \]

      12. log-lowering-log.f64 N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z\right)\right) + a\right) \]

      13. sub-neg N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z\right)\right) + a\right) \]

      14. metadata-eval N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z\right)\right) + a\right) \]

      15. +-lowering-+.f64 84.3

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + -0.5}, z\right)\right) + a\right) \]

   8. Simplified 84.3%

      \[\leadsto \color{blue}{t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right) + a\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto t + \color{blue}{a} \]
   10. Step-by-step derivation

   11. Simplified 32.7%

       \[\leadsto t + \color{blue}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -4 \cdot 10^{+303}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -20:\\ \;\;\;\;z + t\\ \mathbf{elif}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq \infty:\\ \;\;\;\;t + a\\ \mathbf{else}:\\ \;\;\;\;y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(b - 0.5\right) \cdot \log c\\ t_2 := x \cdot \log y\\ t_3 := \left(\left(\left(\left(t\_2 + z\right) + t\right) + a\right) + t\_1\right) + y \cdot i\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+292}:\\ \;\;\;\;t\_2 + y \cdot i\\ \mathbf{elif}\;t\_3 \leq 200:\\ \;\;\;\;t + \left(a + \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(a + t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (- b 0.5) (log c)))
        (t_2 (* x (log y)))
        (t_3 (+ (+ (+ (+ (+ t_2 z) t) a) t_1) (* y i))))
   (if (<= t_3 -5e+292)
     (+ t_2 (* y i))
     (if (<= t_3 200.0)
       (+ t (+ a (fma (log c) (+ b -0.5) z)))
       (+ (* y i) (+ a t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (b - 0.5) * log(c);
	double t_2 = x * log(y);
	double t_3 = ((((t_2 + z) + t) + a) + t_1) + (y * i);
	double tmp;
	if (t_3 <= -5e+292) {
		tmp = t_2 + (y * i);
	} else if (t_3 <= 200.0) {
		tmp = t + (a + fma(log(c), (b + -0.5), z));
	} else {
		tmp = (y * i) + (a + t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(b - 0.5) * log(c))
	t_2 = Float64(x * log(y))
	t_3 = Float64(Float64(Float64(Float64(Float64(t_2 + z) + t) + a) + t_1) + Float64(y * i))
	tmp = 0.0
	if (t_3 <= -5e+292)
		tmp = Float64(t_2 + Float64(y * i));
	elseif (t_3 <= 200.0)
		tmp = Float64(t + Float64(a + fma(log(c), Float64(b + -0.5), z)));
	else
		tmp = Float64(Float64(y * i) + Float64(a + t_1));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(t$95$2 + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+292], N[(t$95$2 + N[(y * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 200.0], N[(t + N[(a + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * i), $MachinePrecision] + N[(a + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -4.9999999999999996e292

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} + y \cdot i \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \log y} + y \cdot i \]

      2. log-lowering-log.f64 85.6

         \[\leadsto x \cdot \color{blue}{\log y} + y \cdot i \]

   5. Simplified 85.6%

      \[\leadsto \color{blue}{x \cdot \log y} + y \cdot i \]

   if -4.9999999999999996e292 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 200

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\color{blue}{z \cdot \left(1 + \frac{x \cdot \log y}{z}\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{z \cdot \left(1 + \frac{x \cdot \log y}{z}\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\left(z \cdot \color{blue}{\left(\frac{x \cdot \log y}{z} + 1\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      3. associate-/l* N/A

         \[\leadsto \left(\left(\left(z \cdot \left(\color{blue}{x \cdot \frac{\log y}{z}} + 1\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\left(z \cdot \color{blue}{\mathsf{fma}\left(x, \frac{\log y}{z}, 1\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(z \cdot \mathsf{fma}\left(x, \color{blue}{\frac{\log y}{z}}, 1\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      6. log-lowering-log.f64 88.2

         \[\leadsto \left(\left(\left(z \cdot \mathsf{fma}\left(x, \frac{\color{blue}{\log y}}{z}, 1\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   5. Simplified 88.2%

      \[\leadsto \left(\left(\left(\color{blue}{z \cdot \mathsf{fma}\left(x, \frac{\log y}{z}, 1\right)} + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{t + \left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{t + \left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto t + \color{blue}{\left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      5. +-commutative N/A

         \[\leadsto t + \left(\color{blue}{\left(\left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right) + z\right)} + a\right) \]

      6. associate-+l+ N/A

         \[\leadsto t + \left(\color{blue}{\left(i \cdot y + \left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)\right)} + a\right) \]

      7. *-commutative N/A

         \[\leadsto t + \left(\left(\color{blue}{y \cdot i} + \left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)\right) + a\right) \]

      8. +-commutative N/A

         \[\leadsto t + \left(\left(y \cdot i + \color{blue}{\left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)}\right) + a\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto t + \left(\color{blue}{\mathsf{fma}\left(y, i, z + \log c \cdot \left(b - \frac{1}{2}\right)\right)} + a\right) \]

      10. +-commutative N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + z}\right) + a\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z\right)}\right) + a\right) \]

      12. log-lowering-log.f64 N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z\right)\right) + a\right) \]

      13. sub-neg N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z\right)\right) + a\right) \]

      14. metadata-eval N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z\right)\right) + a\right) \]

      15. +-lowering-+.f64 86.9

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + -0.5}, z\right)\right) + a\right) \]

   8. Simplified 86.9%

      \[\leadsto \color{blue}{t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right) + a\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto t + \left(\color{blue}{\left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)} + a\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto t + \left(\color{blue}{\left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)} + a\right) \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto t + \left(\color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z\right)} + a\right) \]

       3. log-lowering-log.f64 N/A

          \[\leadsto t + \left(\mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z\right) + a\right) \]

       4. sub-neg N/A

          \[\leadsto t + \left(\mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z\right) + a\right) \]

       5. metadata-eval N/A

          \[\leadsto t + \left(\mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z\right) + a\right) \]

       6. +-lowering-+.f64 78.3

          \[\leadsto t + \left(\mathsf{fma}\left(\log c, \color{blue}{b + -0.5}, z\right) + a\right) \]

   11. Simplified 78.3%

       \[\leadsto t + \left(\color{blue}{\mathsf{fma}\left(\log c, b + -0.5, z\right)} + a\right) \]

   if 200 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \left(\color{blue}{a} + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]
   4. Step-by-step derivation

   5. Simplified 45.1%

      \[\leadsto \left(\color{blue}{a} + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -5 \cdot 10^{+292}:\\ \;\;\;\;x \cdot \log y + y \cdot i\\ \mathbf{elif}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq 200:\\ \;\;\;\;t + \left(a + \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(a + \left(b - 0.5\right) \cdot \log c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+33}:\\ \;\;\;\;t + \mathsf{fma}\left(y, i, z\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+271}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, a\right)\\ \mathbf{else}:\\ \;\;\;\;a + \mathsf{fma}\left(y, i, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 -4e+33)
     (+ t (fma y i z))
     (if (<= t_1 1e+271) (fma x (log y) a) (+ a (fma y i t))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -4e+33) {
		tmp = t + fma(y, i, z);
	} else if (t_1 <= 1e+271) {
		tmp = fma(x, log(y), a);
	} else {
		tmp = a + fma(y, i, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= -4e+33)
		tmp = Float64(t + fma(y, i, z));
	elseif (t_1 <= 1e+271)
		tmp = fma(x, log(y), a);
	else
		tmp = Float64(a + fma(y, i, t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+33], N[(t + N[(y * i + z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+271], N[(x * N[Log[y], $MachinePrecision] + a), $MachinePrecision], N[(a + N[(y * i + t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -3.9999999999999998e33

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)} \]

      2. sub-neg N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{-1}\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right) \cdot \left(-1 \cdot a\right) + -1 \cdot \left(-1 \cdot a\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right)} + -1 \cdot \left(-1 \cdot a\right) \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot a\right) \cdot -1\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}} + -1 \cdot \left(-1 \cdot a\right) \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(-1 \cdot a\right)\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot -1\right) \cdot a\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\color{blue}{1} \cdot a\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      10. *-lft-identity N/A

          \[\leadsto \color{blue}{a} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      11. associate-*r* N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{\left(-1 \cdot -1\right) \cdot a} \]

      12. metadata-eval N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{1} \cdot a \]

      13. *-lft-identity N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{a} \]

   5. Simplified 76.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, t\right)\right)}{a}, a\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \color{blue}{t}}{a}, a\right) \]
   7. Step-by-step derivation

   8. Simplified 60.5%

      \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \color{blue}{t}}{a}, a\right) \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{t + \left(z + i \cdot y\right)} \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{t + \left(z + i \cdot y\right)} \]

       2. +-commutative N/A

          \[\leadsto t + \color{blue}{\left(i \cdot y + z\right)} \]

       3. *-commutative N/A

          \[\leadsto t + \left(\color{blue}{y \cdot i} + z\right) \]

       4. accelerator-lowering-fma.f64 56.0

          \[\leadsto t + \color{blue}{\mathsf{fma}\left(y, i, z\right)} \]

   11. Simplified 56.0%

       \[\leadsto \color{blue}{t + \mathsf{fma}\left(y, i, z\right)} \]

   if -3.9999999999999998e33 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 9.99999999999999953e270

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)} \]

      2. sub-neg N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{-1}\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right) \cdot \left(-1 \cdot a\right) + -1 \cdot \left(-1 \cdot a\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right)} + -1 \cdot \left(-1 \cdot a\right) \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot a\right) \cdot -1\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}} + -1 \cdot \left(-1 \cdot a\right) \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(-1 \cdot a\right)\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot -1\right) \cdot a\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\color{blue}{1} \cdot a\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      10. *-lft-identity N/A

          \[\leadsto \color{blue}{a} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      11. associate-*r* N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{\left(-1 \cdot -1\right) \cdot a} \]

      12. metadata-eval N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{1} \cdot a \]

      13. *-lft-identity N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{a} \]

   5. Simplified 74.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, t\right)\right)}{a}, a\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{x \cdot \log y}{a}}, a\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{x \cdot \log y}{a}}, a\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{x \cdot \log y}}{a}, a\right) \]

      3. log-lowering-log.f64 26.9

         \[\leadsto \mathsf{fma}\left(a, \frac{x \cdot \color{blue}{\log y}}{a}, a\right) \]

   8. Simplified 26.9%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{x \cdot \log y}{a}}, a\right) \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{a + x \cdot \log y} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{x \cdot \log y + a} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, a\right)} \]

       3. log-lowering-log.f64 33.5

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\log y}, a\right) \]

   11. Simplified 33.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, a\right)} \]

   if 9.99999999999999953e270 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)} \]

      2. sub-neg N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{-1}\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right) \cdot \left(-1 \cdot a\right) + -1 \cdot \left(-1 \cdot a\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right)} + -1 \cdot \left(-1 \cdot a\right) \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot a\right) \cdot -1\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}} + -1 \cdot \left(-1 \cdot a\right) \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(-1 \cdot a\right)\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot -1\right) \cdot a\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\color{blue}{1} \cdot a\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      10. *-lft-identity N/A

          \[\leadsto \color{blue}{a} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      11. associate-*r* N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{\left(-1 \cdot -1\right) \cdot a} \]

      12. metadata-eval N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{1} \cdot a \]

      13. *-lft-identity N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{a} \]

   5. Simplified 72.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, t\right)\right)}{a}, a\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \color{blue}{t}}{a}, a\right) \]
   7. Step-by-step derivation

   8. Simplified 57.6%

      \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \color{blue}{t}}{a}, a\right) \]

   9. Taylor expanded in z around 0

      \[\leadsto \color{blue}{a + \left(t + i \cdot y\right)} \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{a + \left(t + i \cdot y\right)} \]

       2. +-commutative N/A

          \[\leadsto a + \color{blue}{\left(i \cdot y + t\right)} \]

       3. *-commutative N/A

          \[\leadsto a + \left(\color{blue}{y \cdot i} + t\right) \]

       4. accelerator-lowering-fma.f64 57.4

          \[\leadsto a + \color{blue}{\mathsf{fma}\left(y, i, t\right)} \]

   11. Simplified 57.4%

       \[\leadsto \color{blue}{a + \mathsf{fma}\left(y, i, t\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 39.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+303}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;t\_1 \leq -20:\\ \;\;\;\;z + t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 -4e+303) (* y i) (if (<= t_1 -20.0) (+ z t) (fma i y a)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -4e+303) {
		tmp = y * i;
	} else if (t_1 <= -20.0) {
		tmp = z + t;
	} else {
		tmp = fma(i, y, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= -4e+303)
		tmp = Float64(y * i);
	elseif (t_1 <= -20.0)
		tmp = Float64(z + t);
	else
		tmp = fma(i, y, a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+303], N[(y * i), $MachinePrecision], If[LessEqual[t$95$1, -20.0], N[(z + t), $MachinePrecision], N[(i * y + a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -4e303

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{i \cdot y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 94.7

         \[\leadsto \color{blue}{i \cdot y} \]

   5. Simplified 94.7%

      \[\leadsto \color{blue}{i \cdot y} \]

   if -4e303 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -20

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\color{blue}{z \cdot \left(1 + \frac{x \cdot \log y}{z}\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{z \cdot \left(1 + \frac{x \cdot \log y}{z}\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\left(z \cdot \color{blue}{\left(\frac{x \cdot \log y}{z} + 1\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      3. associate-/l* N/A

         \[\leadsto \left(\left(\left(z \cdot \left(\color{blue}{x \cdot \frac{\log y}{z}} + 1\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\left(z \cdot \color{blue}{\mathsf{fma}\left(x, \frac{\log y}{z}, 1\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(z \cdot \mathsf{fma}\left(x, \color{blue}{\frac{\log y}{z}}, 1\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      6. log-lowering-log.f64 86.6

         \[\leadsto \left(\left(\left(z \cdot \mathsf{fma}\left(x, \frac{\color{blue}{\log y}}{z}, 1\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   5. Simplified 86.6%

      \[\leadsto \left(\left(\left(\color{blue}{z \cdot \mathsf{fma}\left(x, \frac{\log y}{z}, 1\right)} + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{t + \left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{t + \left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto t + \color{blue}{\left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      5. +-commutative N/A

         \[\leadsto t + \left(\color{blue}{\left(\left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right) + z\right)} + a\right) \]

      6. associate-+l+ N/A

         \[\leadsto t + \left(\color{blue}{\left(i \cdot y + \left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)\right)} + a\right) \]

      7. *-commutative N/A

         \[\leadsto t + \left(\left(\color{blue}{y \cdot i} + \left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)\right) + a\right) \]

      8. +-commutative N/A

         \[\leadsto t + \left(\left(y \cdot i + \color{blue}{\left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)}\right) + a\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto t + \left(\color{blue}{\mathsf{fma}\left(y, i, z + \log c \cdot \left(b - \frac{1}{2}\right)\right)} + a\right) \]

      10. +-commutative N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + z}\right) + a\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z\right)}\right) + a\right) \]

      12. log-lowering-log.f64 N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z\right)\right) + a\right) \]

      13. sub-neg N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z\right)\right) + a\right) \]

      14. metadata-eval N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z\right)\right) + a\right) \]

      15. +-lowering-+.f64 84.5

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + -0.5}, z\right)\right) + a\right) \]

   8. Simplified 84.5%

      \[\leadsto \color{blue}{t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right) + a\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto t + \color{blue}{z} \]
   10. Step-by-step derivation

   11. Simplified 37.6%

       \[\leadsto t + \color{blue}{z} \]

   if -20 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)} \]

      2. sub-neg N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{-1}\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right) \cdot \left(-1 \cdot a\right) + -1 \cdot \left(-1 \cdot a\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right)} + -1 \cdot \left(-1 \cdot a\right) \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot a\right) \cdot -1\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}} + -1 \cdot \left(-1 \cdot a\right) \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(-1 \cdot a\right)\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot -1\right) \cdot a\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\color{blue}{1} \cdot a\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      10. *-lft-identity N/A

          \[\leadsto \color{blue}{a} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      11. associate-*r* N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{\left(-1 \cdot -1\right) \cdot a} \]

      12. metadata-eval N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{1} \cdot a \]

      13. *-lft-identity N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{a} \]

   5. Simplified 71.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, t\right)\right)}{a}, a\right)} \]

   6. Taylor expanded in i around inf

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{i \cdot y}{a}}, a\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{i \cdot y}{a}}, a\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{y \cdot i}}{a}, a\right) \]

      3. *-lowering-*.f64 28.8

         \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{y \cdot i}}{a}, a\right) \]

   8. Simplified 28.8%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{y \cdot i}{a}}, a\right) \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{a + i \cdot y} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{i \cdot y + a} \]

       2. accelerator-lowering-fma.f64 30.1

          \[\leadsto \color{blue}{\mathsf{fma}\left(i, y, a\right)} \]

   11. Simplified 30.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(i, y, a\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -4 \cdot 10^{+303}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -20:\\ \;\;\;\;z + t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+303}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;a + \left(z + t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 -4e+303) (* y i) (if (<= t_1 INFINITY) (+ a (+ z t)) (* y i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -4e+303) {
		tmp = y * i;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = a + (z + t);
	} else {
		tmp = y * i;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
	double tmp;
	if (t_1 <= -4e+303) {
		tmp = y * i;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = a + (z + t);
	} else {
		tmp = y * i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)
	tmp = 0
	if t_1 <= -4e+303:
		tmp = y * i
	elif t_1 <= math.inf:
		tmp = a + (z + t)
	else:
		tmp = y * i
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= -4e+303)
		tmp = Float64(y * i);
	elseif (t_1 <= Inf)
		tmp = Float64(a + Float64(z + t));
	else
		tmp = Float64(y * i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	tmp = 0.0;
	if (t_1 <= -4e+303)
		tmp = y * i;
	elseif (t_1 <= Inf)
		tmp = a + (z + t);
	else
		tmp = y * i;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+303], N[(y * i), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(a + N[(z + t), $MachinePrecision]), $MachinePrecision], N[(y * i), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -4e303 or +inf.0 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{i \cdot y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 94.7

         \[\leadsto \color{blue}{i \cdot y} \]

   5. Simplified 94.7%

      \[\leadsto \color{blue}{i \cdot y} \]

   if -4e303 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < +inf.0

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)} \]

      2. sub-neg N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{-1}\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right) \cdot \left(-1 \cdot a\right) + -1 \cdot \left(-1 \cdot a\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right)} + -1 \cdot \left(-1 \cdot a\right) \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot a\right) \cdot -1\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}} + -1 \cdot \left(-1 \cdot a\right) \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(-1 \cdot a\right)\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot -1\right) \cdot a\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\color{blue}{1} \cdot a\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      10. *-lft-identity N/A

          \[\leadsto \color{blue}{a} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      11. associate-*r* N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{\left(-1 \cdot -1\right) \cdot a} \]

      12. metadata-eval N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{1} \cdot a \]

      13. *-lft-identity N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{a} \]

   5. Simplified 73.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, t\right)\right)}{a}, a\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \color{blue}{t}}{a}, a\right) \]
   7. Step-by-step derivation

   8. Simplified 51.7%

      \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \color{blue}{t}}{a}, a\right) \]

   9. Taylor expanded in i around 0

      \[\leadsto \color{blue}{a + \left(t + z\right)} \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{a + \left(t + z\right)} \]

       2. +-lowering-+.f64 53.3

          \[\leadsto a + \color{blue}{\left(t + z\right)} \]

   11. Simplified 53.3%

       \[\leadsto \color{blue}{a + \left(t + z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -4 \cdot 10^{+303}:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq \infty:\\ \;\;\;\;a + \left(z + t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(b - 0.5\right) \cdot \log c\\ \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + t\_1\right) + y \cdot i \leq 200:\\ \;\;\;\;y \cdot i + \left(z + t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(a + t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (- b 0.5) (log c))))
   (if (<= (+ (+ (+ (+ (+ (* x (log y)) z) t) a) t_1) (* y i)) 200.0)
     (+ (* y i) (+ z t_1))
     (+ (* y i) (+ a t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (b - 0.5) * log(c);
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + t_1) + (y * i)) <= 200.0) {
		tmp = (y * i) + (z + t_1);
	} else {
		tmp = (y * i) + (a + t_1);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (b - 0.5d0) * log(c)
    if (((((((x * log(y)) + z) + t) + a) + t_1) + (y * i)) <= 200.0d0) then
        tmp = (y * i) + (z + t_1)
    else
        tmp = (y * i) + (a + t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (b - 0.5) * Math.log(c);
	double tmp;
	if (((((((x * Math.log(y)) + z) + t) + a) + t_1) + (y * i)) <= 200.0) {
		tmp = (y * i) + (z + t_1);
	} else {
		tmp = (y * i) + (a + t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (b - 0.5) * math.log(c)
	tmp = 0
	if ((((((x * math.log(y)) + z) + t) + a) + t_1) + (y * i)) <= 200.0:
		tmp = (y * i) + (z + t_1)
	else:
		tmp = (y * i) + (a + t_1)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(b - 0.5) * log(c))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + t_1) + Float64(y * i)) <= 200.0)
		tmp = Float64(Float64(y * i) + Float64(z + t_1));
	else
		tmp = Float64(Float64(y * i) + Float64(a + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (b - 0.5) * log(c);
	tmp = 0.0;
	if (((((((x * log(y)) + z) + t) + a) + t_1) + (y * i)) <= 200.0)
		tmp = (y * i) + (z + t_1);
	else
		tmp = (y * i) + (a + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], 200.0], N[(N[(y * i), $MachinePrecision] + N[(z + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(y * i), $MachinePrecision] + N[(a + t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 200

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\color{blue}{z} + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]
   4. Step-by-step derivation

   5. Simplified 54.8%

      \[\leadsto \left(\color{blue}{z} + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   if 200 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \left(\color{blue}{a} + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]
   4. Step-by-step derivation

   5. Simplified 45.1%

      \[\leadsto \left(\color{blue}{a} + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq 200:\\ \;\;\;\;y \cdot i + \left(z + \left(b - 0.5\right) \cdot \log c\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(a + \left(b - 0.5\right) \cdot \log c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -20:\\ \;\;\;\;t + \mathsf{fma}\left(y, i, z\right)\\ \mathbf{else}:\\ \;\;\;\;a + \mathsf{fma}\left(y, i, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
      -20.0)
   (+ t (fma y i z))
   (+ a (fma y i t))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -20.0) {
		tmp = t + fma(y, i, z);
	} else {
		tmp = a + fma(y, i, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= -20.0)
		tmp = Float64(t + fma(y, i, z));
	else
		tmp = Float64(a + fma(y, i, t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], -20.0], N[(t + N[(y * i + z), $MachinePrecision]), $MachinePrecision], N[(a + N[(y * i + t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -20

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)} \]

      2. sub-neg N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{-1}\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right) \cdot \left(-1 \cdot a\right) + -1 \cdot \left(-1 \cdot a\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right)} + -1 \cdot \left(-1 \cdot a\right) \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot a\right) \cdot -1\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}} + -1 \cdot \left(-1 \cdot a\right) \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(-1 \cdot a\right)\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot -1\right) \cdot a\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\color{blue}{1} \cdot a\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      10. *-lft-identity N/A

          \[\leadsto \color{blue}{a} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      11. associate-*r* N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{\left(-1 \cdot -1\right) \cdot a} \]

      12. metadata-eval N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{1} \cdot a \]

      13. *-lft-identity N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{a} \]

   5. Simplified 77.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, t\right)\right)}{a}, a\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \color{blue}{t}}{a}, a\right) \]
   7. Step-by-step derivation

   8. Simplified 57.0%

      \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \color{blue}{t}}{a}, a\right) \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{t + \left(z + i \cdot y\right)} \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{t + \left(z + i \cdot y\right)} \]

       2. +-commutative N/A

          \[\leadsto t + \color{blue}{\left(i \cdot y + z\right)} \]

       3. *-commutative N/A

          \[\leadsto t + \left(\color{blue}{y \cdot i} + z\right) \]

       4. accelerator-lowering-fma.f64 52.9

          \[\leadsto t + \color{blue}{\mathsf{fma}\left(y, i, z\right)} \]

   11. Simplified 52.9%

       \[\leadsto \color{blue}{t + \mathsf{fma}\left(y, i, z\right)} \]

   if -20 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)} \]

      2. sub-neg N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{-1}\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right) \cdot \left(-1 \cdot a\right) + -1 \cdot \left(-1 \cdot a\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right)} + -1 \cdot \left(-1 \cdot a\right) \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot a\right) \cdot -1\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}} + -1 \cdot \left(-1 \cdot a\right) \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(-1 \cdot a\right)\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot -1\right) \cdot a\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\color{blue}{1} \cdot a\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      10. *-lft-identity N/A

          \[\leadsto \color{blue}{a} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      11. associate-*r* N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{\left(-1 \cdot -1\right) \cdot a} \]

      12. metadata-eval N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{1} \cdot a \]

      13. *-lft-identity N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{a} \]

   5. Simplified 71.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, t\right)\right)}{a}, a\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \color{blue}{t}}{a}, a\right) \]
   7. Step-by-step derivation

   8. Simplified 52.4%

      \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \color{blue}{t}}{a}, a\right) \]

   9. Taylor expanded in z around 0

      \[\leadsto \color{blue}{a + \left(t + i \cdot y\right)} \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{a + \left(t + i \cdot y\right)} \]

       2. +-commutative N/A

          \[\leadsto a + \color{blue}{\left(i \cdot y + t\right)} \]

       3. *-commutative N/A

          \[\leadsto a + \left(\color{blue}{y \cdot i} + t\right) \]

       4. accelerator-lowering-fma.f64 48.5

          \[\leadsto a + \color{blue}{\mathsf{fma}\left(y, i, t\right)} \]

   11. Simplified 48.5%

       \[\leadsto \color{blue}{a + \mathsf{fma}\left(y, i, t\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 46.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -20:\\ \;\;\;\;z + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;a + \mathsf{fma}\left(y, i, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
      -20.0)
   (+ z (* y i))
   (+ a (fma y i t))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -20.0) {
		tmp = z + (y * i);
	} else {
		tmp = a + fma(y, i, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= -20.0)
		tmp = Float64(z + Float64(y * i));
	else
		tmp = Float64(a + fma(y, i, t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], -20.0], N[(z + N[(y * i), $MachinePrecision]), $MachinePrecision], N[(a + N[(y * i + t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -20

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z} + y \cdot i \]
   4. Step-by-step derivation

   5. Simplified 36.6%

      \[\leadsto \color{blue}{z} + y \cdot i \]

   if -20 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)} \]

      2. sub-neg N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{-1}\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right) \cdot \left(-1 \cdot a\right) + -1 \cdot \left(-1 \cdot a\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right)} + -1 \cdot \left(-1 \cdot a\right) \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot a\right) \cdot -1\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}} + -1 \cdot \left(-1 \cdot a\right) \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(-1 \cdot a\right)\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot -1\right) \cdot a\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\color{blue}{1} \cdot a\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      10. *-lft-identity N/A

          \[\leadsto \color{blue}{a} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      11. associate-*r* N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{\left(-1 \cdot -1\right) \cdot a} \]

      12. metadata-eval N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{1} \cdot a \]

      13. *-lft-identity N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{a} \]

   5. Simplified 71.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, t\right)\right)}{a}, a\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \color{blue}{t}}{a}, a\right) \]
   7. Step-by-step derivation

   8. Simplified 52.4%

      \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \color{blue}{t}}{a}, a\right) \]

   9. Taylor expanded in z around 0

      \[\leadsto \color{blue}{a + \left(t + i \cdot y\right)} \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{a + \left(t + i \cdot y\right)} \]

       2. +-commutative N/A

          \[\leadsto a + \color{blue}{\left(i \cdot y + t\right)} \]

       3. *-commutative N/A

          \[\leadsto a + \left(\color{blue}{y \cdot i} + t\right) \]

       4. accelerator-lowering-fma.f64 48.5

          \[\leadsto a + \color{blue}{\mathsf{fma}\left(y, i, t\right)} \]

   11. Simplified 48.5%

       \[\leadsto \color{blue}{a + \mathsf{fma}\left(y, i, t\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 38.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -20:\\ \;\;\;\;z + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
      -20.0)
   (+ z (* y i))
   (fma i y a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -20.0) {
		tmp = z + (y * i);
	} else {
		tmp = fma(i, y, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= -20.0)
		tmp = Float64(z + Float64(y * i));
	else
		tmp = fma(i, y, a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], -20.0], N[(z + N[(y * i), $MachinePrecision]), $MachinePrecision], N[(i * y + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -20

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z} + y \cdot i \]
   4. Step-by-step derivation

   5. Simplified 36.6%

      \[\leadsto \color{blue}{z} + y \cdot i \]

   if -20 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)} \]

      2. sub-neg N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{-1}\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right) \cdot \left(-1 \cdot a\right) + -1 \cdot \left(-1 \cdot a\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right)} + -1 \cdot \left(-1 \cdot a\right) \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot a\right) \cdot -1\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}} + -1 \cdot \left(-1 \cdot a\right) \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(-1 \cdot a\right)\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot -1\right) \cdot a\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\color{blue}{1} \cdot a\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      10. *-lft-identity N/A

          \[\leadsto \color{blue}{a} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      11. associate-*r* N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{\left(-1 \cdot -1\right) \cdot a} \]

      12. metadata-eval N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{1} \cdot a \]

      13. *-lft-identity N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{a} \]

   5. Simplified 71.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, t\right)\right)}{a}, a\right)} \]

   6. Taylor expanded in i around inf

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{i \cdot y}{a}}, a\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{i \cdot y}{a}}, a\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{y \cdot i}}{a}, a\right) \]

      3. *-lowering-*.f64 28.8

         \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{y \cdot i}}{a}, a\right) \]

   8. Simplified 28.8%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{y \cdot i}{a}}, a\right) \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{a + i \cdot y} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{i \cdot y + a} \]

       2. accelerator-lowering-fma.f64 30.1

          \[\leadsto \color{blue}{\mathsf{fma}\left(i, y, a\right)} \]

   11. Simplified 30.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(i, y, a\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 30.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -20:\\ \;\;\;\;z + t\\ \mathbf{else}:\\ \;\;\;\;t + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
      -20.0)
   (+ z t)
   (+ t a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -20.0) {
		tmp = z + t;
	} else {
		tmp = t + a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)) <= (-20.0d0)) then
        tmp = z + t
    else
        tmp = t + a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i)) <= -20.0) {
		tmp = z + t;
	} else {
		tmp = t + a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)) <= -20.0:
		tmp = z + t
	else:
		tmp = t + a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= -20.0)
		tmp = Float64(z + t);
	else
		tmp = Float64(t + a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -20.0)
		tmp = z + t;
	else
		tmp = t + a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], -20.0], N[(z + t), $MachinePrecision], N[(t + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -20

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\color{blue}{z \cdot \left(1 + \frac{x \cdot \log y}{z}\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{z \cdot \left(1 + \frac{x \cdot \log y}{z}\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\left(z \cdot \color{blue}{\left(\frac{x \cdot \log y}{z} + 1\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      3. associate-/l* N/A

         \[\leadsto \left(\left(\left(z \cdot \left(\color{blue}{x \cdot \frac{\log y}{z}} + 1\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\left(z \cdot \color{blue}{\mathsf{fma}\left(x, \frac{\log y}{z}, 1\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(z \cdot \mathsf{fma}\left(x, \color{blue}{\frac{\log y}{z}}, 1\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      6. log-lowering-log.f64 86.9

         \[\leadsto \left(\left(\left(z \cdot \mathsf{fma}\left(x, \frac{\color{blue}{\log y}}{z}, 1\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   5. Simplified 86.9%

      \[\leadsto \left(\left(\left(\color{blue}{z \cdot \mathsf{fma}\left(x, \frac{\log y}{z}, 1\right)} + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{t + \left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{t + \left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto t + \color{blue}{\left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      5. +-commutative N/A

         \[\leadsto t + \left(\color{blue}{\left(\left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right) + z\right)} + a\right) \]

      6. associate-+l+ N/A

         \[\leadsto t + \left(\color{blue}{\left(i \cdot y + \left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)\right)} + a\right) \]

      7. *-commutative N/A

         \[\leadsto t + \left(\left(\color{blue}{y \cdot i} + \left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)\right) + a\right) \]

      8. +-commutative N/A

         \[\leadsto t + \left(\left(y \cdot i + \color{blue}{\left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)}\right) + a\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto t + \left(\color{blue}{\mathsf{fma}\left(y, i, z + \log c \cdot \left(b - \frac{1}{2}\right)\right)} + a\right) \]

      10. +-commutative N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + z}\right) + a\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z\right)}\right) + a\right) \]

      12. log-lowering-log.f64 N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z\right)\right) + a\right) \]

      13. sub-neg N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z\right)\right) + a\right) \]

      14. metadata-eval N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z\right)\right) + a\right) \]

      15. +-lowering-+.f64 85.8

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + -0.5}, z\right)\right) + a\right) \]

   8. Simplified 85.8%

      \[\leadsto \color{blue}{t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right) + a\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto t + \color{blue}{z} \]
   10. Step-by-step derivation

   11. Simplified 32.7%

       \[\leadsto t + \color{blue}{z} \]

   if -20 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\color{blue}{z \cdot \left(1 + \frac{x \cdot \log y}{z}\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{z \cdot \left(1 + \frac{x \cdot \log y}{z}\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\left(z \cdot \color{blue}{\left(\frac{x \cdot \log y}{z} + 1\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      3. associate-/l* N/A

         \[\leadsto \left(\left(\left(z \cdot \left(\color{blue}{x \cdot \frac{\log y}{z}} + 1\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\left(z \cdot \color{blue}{\mathsf{fma}\left(x, \frac{\log y}{z}, 1\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(z \cdot \mathsf{fma}\left(x, \color{blue}{\frac{\log y}{z}}, 1\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      6. log-lowering-log.f64 90.8

         \[\leadsto \left(\left(\left(z \cdot \mathsf{fma}\left(x, \frac{\color{blue}{\log y}}{z}, 1\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   5. Simplified 90.8%

      \[\leadsto \left(\left(\left(\color{blue}{z \cdot \mathsf{fma}\left(x, \frac{\log y}{z}, 1\right)} + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{t + \left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{t + \left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto t + \color{blue}{\left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      5. +-commutative N/A

         \[\leadsto t + \left(\color{blue}{\left(\left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right) + z\right)} + a\right) \]

      6. associate-+l+ N/A

         \[\leadsto t + \left(\color{blue}{\left(i \cdot y + \left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)\right)} + a\right) \]

      7. *-commutative N/A

         \[\leadsto t + \left(\left(\color{blue}{y \cdot i} + \left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)\right) + a\right) \]

      8. +-commutative N/A

         \[\leadsto t + \left(\left(y \cdot i + \color{blue}{\left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)}\right) + a\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto t + \left(\color{blue}{\mathsf{fma}\left(y, i, z + \log c \cdot \left(b - \frac{1}{2}\right)\right)} + a\right) \]

      10. +-commutative N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + z}\right) + a\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z\right)}\right) + a\right) \]

      12. log-lowering-log.f64 N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z\right)\right) + a\right) \]

      13. sub-neg N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z\right)\right) + a\right) \]

      14. metadata-eval N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z\right)\right) + a\right) \]

      15. +-lowering-+.f64 84.3

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + -0.5}, z\right)\right) + a\right) \]

   8. Simplified 84.3%

      \[\leadsto \color{blue}{t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right) + a\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto t + \color{blue}{a} \]
   10. Step-by-step derivation

   11. Simplified 32.7%

       \[\leadsto t + \color{blue}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -20:\\ \;\;\;\;z + t\\ \mathbf{else}:\\ \;\;\;\;t + a\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 23.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -20:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;t + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
      -20.0)
   z
   (+ t a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -20.0) {
		tmp = z;
	} else {
		tmp = t + a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)) <= (-20.0d0)) then
        tmp = z
    else
        tmp = t + a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i)) <= -20.0) {
		tmp = z;
	} else {
		tmp = t + a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)) <= -20.0:
		tmp = z
	else:
		tmp = t + a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= -20.0)
		tmp = z;
	else
		tmp = Float64(t + a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -20.0)
		tmp = z;
	else
		tmp = t + a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], -20.0], z, N[(t + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -20

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z} \]
   4. Step-by-step derivation

   5. Simplified 16.5%

      \[\leadsto \color{blue}{z} \]

   if -20 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\color{blue}{z \cdot \left(1 + \frac{x \cdot \log y}{z}\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{z \cdot \left(1 + \frac{x \cdot \log y}{z}\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\left(z \cdot \color{blue}{\left(\frac{x \cdot \log y}{z} + 1\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      3. associate-/l* N/A

         \[\leadsto \left(\left(\left(z \cdot \left(\color{blue}{x \cdot \frac{\log y}{z}} + 1\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\left(z \cdot \color{blue}{\mathsf{fma}\left(x, \frac{\log y}{z}, 1\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(z \cdot \mathsf{fma}\left(x, \color{blue}{\frac{\log y}{z}}, 1\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      6. log-lowering-log.f64 90.8

         \[\leadsto \left(\left(\left(z \cdot \mathsf{fma}\left(x, \frac{\color{blue}{\log y}}{z}, 1\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   5. Simplified 90.8%

      \[\leadsto \left(\left(\left(\color{blue}{z \cdot \mathsf{fma}\left(x, \frac{\log y}{z}, 1\right)} + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{t + \left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{t + \left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto t + \color{blue}{\left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      5. +-commutative N/A

         \[\leadsto t + \left(\color{blue}{\left(\left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right) + z\right)} + a\right) \]

      6. associate-+l+ N/A

         \[\leadsto t + \left(\color{blue}{\left(i \cdot y + \left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)\right)} + a\right) \]

      7. *-commutative N/A

         \[\leadsto t + \left(\left(\color{blue}{y \cdot i} + \left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)\right) + a\right) \]

      8. +-commutative N/A

         \[\leadsto t + \left(\left(y \cdot i + \color{blue}{\left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)}\right) + a\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto t + \left(\color{blue}{\mathsf{fma}\left(y, i, z + \log c \cdot \left(b - \frac{1}{2}\right)\right)} + a\right) \]

      10. +-commutative N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + z}\right) + a\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z\right)}\right) + a\right) \]

      12. log-lowering-log.f64 N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z\right)\right) + a\right) \]

      13. sub-neg N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z\right)\right) + a\right) \]

      14. metadata-eval N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z\right)\right) + a\right) \]

      15. +-lowering-+.f64 84.3

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + -0.5}, z\right)\right) + a\right) \]

   8. Simplified 84.3%

      \[\leadsto \color{blue}{t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right) + a\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto t + \color{blue}{a} \]
   10. Step-by-step derivation

   11. Simplified 32.7%

       \[\leadsto t + \color{blue}{a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 16.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -20:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
      -20.0)
   z
   a))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -20.0) {
		tmp = z;
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)) <= (-20.0d0)) then
        tmp = z
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i)) <= -20.0) {
		tmp = z;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)) <= -20.0:
		tmp = z
	else:
		tmp = a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= -20.0)
		tmp = z;
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -20.0)
		tmp = z;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], -20.0], z, a]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -20

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z} \]
   4. Step-by-step derivation

   5. Simplified 16.5%

      \[\leadsto \color{blue}{z} \]

   if -20 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a} \]
   4. Step-by-step derivation

   5. Simplified 14.1%

      \[\leadsto \color{blue}{a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 91.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\log c, b + -0.5, z\right)\\ \mathbf{if}\;y \leq 8.4 \cdot 10^{+37}:\\ \;\;\;\;a + \left(t\_1 + \mathsf{fma}\left(x, \log y, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t + \left(a + \mathsf{fma}\left(y, i, t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (log c) (+ b -0.5) z)))
   (if (<= y 8.4e+37)
     (+ a (+ t_1 (fma x (log y) t)))
     (+ t (+ a (fma y i t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(log(c), (b + -0.5), z);
	double tmp;
	if (y <= 8.4e+37) {
		tmp = a + (t_1 + fma(x, log(y), t));
	} else {
		tmp = t + (a + fma(y, i, t_1));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(log(c), Float64(b + -0.5), z)
	tmp = 0.0
	if (y <= 8.4e+37)
		tmp = Float64(a + Float64(t_1 + fma(x, log(y), t)));
	else
		tmp = Float64(t + Float64(a + fma(y, i, t_1)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + z), $MachinePrecision]}, If[LessEqual[y, 8.4e+37], N[(a + N[(t$95$1 + N[(x * N[Log[y], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t + N[(a + N[(y * i + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 8.4000000000000004e37

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{a + \left(t + \left(z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto a + \color{blue}{\left(\left(t + z\right) + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      3. cancel-sign-sub N/A

         \[\leadsto a + \left(\left(t + z\right) + \color{blue}{\left(x \cdot \log y - \left(\mathsf{neg}\left(\log c\right)\right) \cdot \left(b - \frac{1}{2}\right)\right)}\right) \]

      4. log-rec N/A

         \[\leadsto a + \left(\left(t + z\right) + \left(x \cdot \log y - \color{blue}{\log \left(\frac{1}{c}\right)} \cdot \left(b - \frac{1}{2}\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto a + \left(\left(t + z\right) + \color{blue}{\left(x \cdot \log y + \left(\mathsf{neg}\left(\log \left(\frac{1}{c}\right) \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}\right) \]

      6. mul-1-neg N/A

         \[\leadsto a + \left(\left(t + z\right) + \left(x \cdot \log y + \color{blue}{-1 \cdot \left(\log \left(\frac{1}{c}\right) \cdot \left(b - \frac{1}{2}\right)\right)}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto a + \left(\left(t + z\right) + \color{blue}{\left(-1 \cdot \left(\log \left(\frac{1}{c}\right) \cdot \left(b - \frac{1}{2}\right)\right) + x \cdot \log y\right)}\right) \]

      8. associate-+r+ N/A

         \[\leadsto a + \color{blue}{\left(t + \left(z + \left(-1 \cdot \left(\log \left(\frac{1}{c}\right) \cdot \left(b - \frac{1}{2}\right)\right) + x \cdot \log y\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto a + \color{blue}{\left(\left(z + \left(-1 \cdot \left(\log \left(\frac{1}{c}\right) \cdot \left(b - \frac{1}{2}\right)\right) + x \cdot \log y\right)\right) + t\right)} \]

      10. associate-+r+ N/A

          \[\leadsto a + \left(\color{blue}{\left(\left(z + -1 \cdot \left(\log \left(\frac{1}{c}\right) \cdot \left(b - \frac{1}{2}\right)\right)\right) + x \cdot \log y\right)} + t\right) \]

      11. associate-+l+ N/A

          \[\leadsto a + \color{blue}{\left(\left(z + -1 \cdot \left(\log \left(\frac{1}{c}\right) \cdot \left(b - \frac{1}{2}\right)\right)\right) + \left(x \cdot \log y + t\right)\right)} \]

   5. Simplified 99.2%

      \[\leadsto \color{blue}{a + \left(\mathsf{fma}\left(\log c, b + -0.5, z\right) + \mathsf{fma}\left(x, \log y, t\right)\right)} \]

   if 8.4000000000000004e37 < y

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\color{blue}{z \cdot \left(1 + \frac{x \cdot \log y}{z}\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{z \cdot \left(1 + \frac{x \cdot \log y}{z}\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\left(z \cdot \color{blue}{\left(\frac{x \cdot \log y}{z} + 1\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      3. associate-/l* N/A

         \[\leadsto \left(\left(\left(z \cdot \left(\color{blue}{x \cdot \frac{\log y}{z}} + 1\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\left(z \cdot \color{blue}{\mathsf{fma}\left(x, \frac{\log y}{z}, 1\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(z \cdot \mathsf{fma}\left(x, \color{blue}{\frac{\log y}{z}}, 1\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      6. log-lowering-log.f64 88.1

         \[\leadsto \left(\left(\left(z \cdot \mathsf{fma}\left(x, \frac{\color{blue}{\log y}}{z}, 1\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   5. Simplified 88.1%

      \[\leadsto \left(\left(\left(\color{blue}{z \cdot \mathsf{fma}\left(x, \frac{\log y}{z}, 1\right)} + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{t + \left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{t + \left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto t + \color{blue}{\left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      5. +-commutative N/A

         \[\leadsto t + \left(\color{blue}{\left(\left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right) + z\right)} + a\right) \]

      6. associate-+l+ N/A

         \[\leadsto t + \left(\color{blue}{\left(i \cdot y + \left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)\right)} + a\right) \]

      7. *-commutative N/A

         \[\leadsto t + \left(\left(\color{blue}{y \cdot i} + \left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)\right) + a\right) \]

      8. +-commutative N/A

         \[\leadsto t + \left(\left(y \cdot i + \color{blue}{\left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)}\right) + a\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto t + \left(\color{blue}{\mathsf{fma}\left(y, i, z + \log c \cdot \left(b - \frac{1}{2}\right)\right)} + a\right) \]

      10. +-commutative N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + z}\right) + a\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z\right)}\right) + a\right) \]

      12. log-lowering-log.f64 N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z\right)\right) + a\right) \]

      13. sub-neg N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z\right)\right) + a\right) \]

      14. metadata-eval N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z\right)\right) + a\right) \]

      15. +-lowering-+.f64 89.4

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + -0.5}, z\right)\right) + a\right) \]

   8. Simplified 89.4%

      \[\leadsto \color{blue}{t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right) + a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.4 \cdot 10^{+37}:\\ \;\;\;\;a + \left(\mathsf{fma}\left(\log c, b + -0.5, z\right) + \mathsf{fma}\left(x, \log y, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t + \left(a + \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 90.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+198}:\\ \;\;\;\;x \cdot \log y + y \cdot i\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+227}:\\ \;\;\;\;t + \left(a + \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log y + \frac{z}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x -5.2e+198)
   (+ (* x (log y)) (* y i))
   (if (<= x 8.5e+227)
     (+ t (+ a (fma y i (fma (log c) (+ b -0.5) z))))
     (* x (+ (log y) (/ z x))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (x <= -5.2e+198) {
		tmp = (x * log(y)) + (y * i);
	} else if (x <= 8.5e+227) {
		tmp = t + (a + fma(y, i, fma(log(c), (b + -0.5), z)));
	} else {
		tmp = x * (log(y) + (z / x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= -5.2e+198)
		tmp = Float64(Float64(x * log(y)) + Float64(y * i));
	elseif (x <= 8.5e+227)
		tmp = Float64(t + Float64(a + fma(y, i, fma(log(c), Float64(b + -0.5), z))));
	else
		tmp = Float64(x * Float64(log(y) + Float64(z / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, -5.2e+198], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.5e+227], N[(t + N[(a + N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[Log[y], $MachinePrecision] + N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.19999999999999961e198

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} + y \cdot i \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \log y} + y \cdot i \]

      2. log-lowering-log.f64 92.1

         \[\leadsto x \cdot \color{blue}{\log y} + y \cdot i \]

   5. Simplified 92.1%

      \[\leadsto \color{blue}{x \cdot \log y} + y \cdot i \]

   if -5.19999999999999961e198 < x < 8.4999999999999995e227

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\color{blue}{z \cdot \left(1 + \frac{x \cdot \log y}{z}\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{z \cdot \left(1 + \frac{x \cdot \log y}{z}\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\left(z \cdot \color{blue}{\left(\frac{x \cdot \log y}{z} + 1\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      3. associate-/l* N/A

         \[\leadsto \left(\left(\left(z \cdot \left(\color{blue}{x \cdot \frac{\log y}{z}} + 1\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\left(z \cdot \color{blue}{\mathsf{fma}\left(x, \frac{\log y}{z}, 1\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(z \cdot \mathsf{fma}\left(x, \color{blue}{\frac{\log y}{z}}, 1\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      6. log-lowering-log.f64 93.5

         \[\leadsto \left(\left(\left(z \cdot \mathsf{fma}\left(x, \frac{\color{blue}{\log y}}{z}, 1\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   5. Simplified 93.5%

      \[\leadsto \left(\left(\left(\color{blue}{z \cdot \mathsf{fma}\left(x, \frac{\log y}{z}, 1\right)} + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{t + \left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{t + \left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto t + \color{blue}{\left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      5. +-commutative N/A

         \[\leadsto t + \left(\color{blue}{\left(\left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right) + z\right)} + a\right) \]

      6. associate-+l+ N/A

         \[\leadsto t + \left(\color{blue}{\left(i \cdot y + \left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)\right)} + a\right) \]

      7. *-commutative N/A

         \[\leadsto t + \left(\left(\color{blue}{y \cdot i} + \left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)\right) + a\right) \]

      8. +-commutative N/A

         \[\leadsto t + \left(\left(y \cdot i + \color{blue}{\left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)}\right) + a\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto t + \left(\color{blue}{\mathsf{fma}\left(y, i, z + \log c \cdot \left(b - \frac{1}{2}\right)\right)} + a\right) \]

      10. +-commutative N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + z}\right) + a\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z\right)}\right) + a\right) \]

      12. log-lowering-log.f64 N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z\right)\right) + a\right) \]

      13. sub-neg N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z\right)\right) + a\right) \]

      14. metadata-eval N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z\right)\right) + a\right) \]

      15. +-lowering-+.f64 93.3

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + -0.5}, z\right)\right) + a\right) \]

   8. Simplified 93.3%

      \[\leadsto \color{blue}{t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right) + a\right)} \]

   if 8.4999999999999995e227 < x

   1. Initial program 99.7%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \log y + -1 \cdot \frac{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{x}\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(-1 \cdot \log y + -1 \cdot \frac{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{x}\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \log y + -1 \cdot \frac{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{x}\right)\right)\right)} \]

      3. distribute-lft-out N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{-1 \cdot \left(\log y + \frac{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{x}\right)}\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\log y + \frac{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{x}\right)\right)\right)}\right)\right) \]

      5. remove-double-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(\log y + \frac{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{x}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\log y + \frac{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{x}\right)} \]

      7. +-lowering-+.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\log y + \frac{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{x}\right)} \]

      8. log-lowering-log.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{\log y} + \frac{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{x}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto x \cdot \left(\log y + \color{blue}{\frac{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{x}}\right) \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \left(\log y + \frac{a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)}{x}\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto x \cdot \left(\log y + \color{blue}{\frac{z}{x}}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 69.3

         \[\leadsto x \cdot \left(\log y + \color{blue}{\frac{z}{x}}\right) \]

   8. Simplified 69.3%

      \[\leadsto x \cdot \left(\log y + \color{blue}{\frac{z}{x}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+198}:\\ \;\;\;\;x \cdot \log y + y \cdot i\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+227}:\\ \;\;\;\;t + \left(a + \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log y + \frac{z}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 90.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+205}:\\ \;\;\;\;x \cdot \log y + y \cdot i\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+228}:\\ \;\;\;\;a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log y + \frac{z}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x -3.6e+205)
   (+ (* x (log y)) (* y i))
   (if (<= x 3.8e+228)
     (+ a (+ (fma i y z) (fma (log c) (+ b -0.5) t)))
     (* x (+ (log y) (/ z x))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (x <= -3.6e+205) {
		tmp = (x * log(y)) + (y * i);
	} else if (x <= 3.8e+228) {
		tmp = a + (fma(i, y, z) + fma(log(c), (b + -0.5), t));
	} else {
		tmp = x * (log(y) + (z / x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= -3.6e+205)
		tmp = Float64(Float64(x * log(y)) + Float64(y * i));
	elseif (x <= 3.8e+228)
		tmp = Float64(a + Float64(fma(i, y, z) + fma(log(c), Float64(b + -0.5), t)));
	else
		tmp = Float64(x * Float64(log(y) + Float64(z / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, -3.6e+205], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.8e+228], N[(a + N[(N[(i * y + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[Log[y], $MachinePrecision] + N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.60000000000000002e205

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} + y \cdot i \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \log y} + y \cdot i \]

      2. log-lowering-log.f64 92.1

         \[\leadsto x \cdot \color{blue}{\log y} + y \cdot i \]

   5. Simplified 92.1%

      \[\leadsto \color{blue}{x \cdot \log y} + y \cdot i \]

   if -3.60000000000000002e205 < x < 3.8000000000000002e228

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto a + \color{blue}{\left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + t\right)} \]

      3. associate-+r+ N/A

         \[\leadsto a + \left(\color{blue}{\left(\left(z + i \cdot y\right) + \log c \cdot \left(b - \frac{1}{2}\right)\right)} + t\right) \]

      4. associate-+l+ N/A

         \[\leadsto a + \color{blue}{\left(\left(z + i \cdot y\right) + \left(\log c \cdot \left(b - \frac{1}{2}\right) + t\right)\right)} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto a + \color{blue}{\left(\left(z + i \cdot y\right) + \left(\log c \cdot \left(b - \frac{1}{2}\right) + t\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto a + \left(\color{blue}{\left(i \cdot y + z\right)} + \left(\log c \cdot \left(b - \frac{1}{2}\right) + t\right)\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto a + \left(\color{blue}{\mathsf{fma}\left(i, y, z\right)} + \left(\log c \cdot \left(b - \frac{1}{2}\right) + t\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto a + \left(\mathsf{fma}\left(i, y, z\right) + \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, t\right)}\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, t\right)\right) \]

      10. sub-neg N/A

          \[\leadsto a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, t\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, t\right)\right) \]

      12. +-lowering-+.f64 93.3

          \[\leadsto a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, \color{blue}{b + -0.5}, t\right)\right) \]

   5. Simplified 93.3%

      \[\leadsto \color{blue}{a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)} \]

   if 3.8000000000000002e228 < x

   1. Initial program 99.7%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \log y + -1 \cdot \frac{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{x}\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(-1 \cdot \log y + -1 \cdot \frac{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{x}\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \log y + -1 \cdot \frac{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{x}\right)\right)\right)} \]

      3. distribute-lft-out N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{-1 \cdot \left(\log y + \frac{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{x}\right)}\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\log y + \frac{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{x}\right)\right)\right)}\right)\right) \]

      5. remove-double-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(\log y + \frac{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{x}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\log y + \frac{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{x}\right)} \]

      7. +-lowering-+.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\log y + \frac{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{x}\right)} \]

      8. log-lowering-log.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{\log y} + \frac{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{x}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto x \cdot \left(\log y + \color{blue}{\frac{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{x}}\right) \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \left(\log y + \frac{a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)}{x}\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto x \cdot \left(\log y + \color{blue}{\frac{z}{x}}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 69.3

         \[\leadsto x \cdot \left(\log y + \color{blue}{\frac{z}{x}}\right) \]

   8. Simplified 69.3%

      \[\leadsto x \cdot \left(\log y + \color{blue}{\frac{z}{x}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 75.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(y + \frac{a + \left(z + t\right)}{i}\right)\\ \mathbf{if}\;i \leq -9.6 \cdot 10^{+62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 2 \cdot 10^{+46}:\\ \;\;\;\;t + \left(a + \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* i (+ y (/ (+ a (+ z t)) i)))))
   (if (<= i -9.6e+62)
     t_1
     (if (<= i 2e+46) (+ t (+ a (fma (log c) (+ b -0.5) z))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = i * (y + ((a + (z + t)) / i));
	double tmp;
	if (i <= -9.6e+62) {
		tmp = t_1;
	} else if (i <= 2e+46) {
		tmp = t + (a + fma(log(c), (b + -0.5), z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(i * Float64(y + Float64(Float64(a + Float64(z + t)) / i)))
	tmp = 0.0
	if (i <= -9.6e+62)
		tmp = t_1;
	elseif (i <= 2e+46)
		tmp = Float64(t + Float64(a + fma(log(c), Float64(b + -0.5), z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * N[(y + N[(N[(a + N[(z + t), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -9.6e+62], t$95$1, If[LessEqual[i, 2e+46], N[(t + N[(a + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if i < -9.6e62 or 2e46 < i

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)} \]

      2. sub-neg N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{-1}\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right) \cdot \left(-1 \cdot a\right) + -1 \cdot \left(-1 \cdot a\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right)} + -1 \cdot \left(-1 \cdot a\right) \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot a\right) \cdot -1\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}} + -1 \cdot \left(-1 \cdot a\right) \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(-1 \cdot a\right)\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot -1\right) \cdot a\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\color{blue}{1} \cdot a\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      10. *-lft-identity N/A

          \[\leadsto \color{blue}{a} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      11. associate-*r* N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{\left(-1 \cdot -1\right) \cdot a} \]

      12. metadata-eval N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{1} \cdot a \]

      13. *-lft-identity N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{a} \]

   5. Simplified 82.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, t\right)\right)}{a}, a\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \color{blue}{t}}{a}, a\right) \]
   7. Step-by-step derivation

   8. Simplified 69.0%

      \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \color{blue}{t}}{a}, a\right) \]

   9. Taylor expanded in i around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(i \cdot \left(-1 \cdot y + -1 \cdot \frac{a + \left(t + z\right)}{i}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(-1 \cdot i\right) \cdot \left(-1 \cdot y + -1 \cdot \frac{a + \left(t + z\right)}{i}\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(-1 \cdot i\right) \cdot \left(-1 \cdot y + -1 \cdot \frac{a + \left(t + z\right)}{i}\right)} \]

       3. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(i\right)\right)} \cdot \left(-1 \cdot y + -1 \cdot \frac{a + \left(t + z\right)}{i}\right) \]

       4. neg-sub0 N/A

          \[\leadsto \color{blue}{\left(0 - i\right)} \cdot \left(-1 \cdot y + -1 \cdot \frac{a + \left(t + z\right)}{i}\right) \]

       5. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(0 - i\right)} \cdot \left(-1 \cdot y + -1 \cdot \frac{a + \left(t + z\right)}{i}\right) \]

       6. mul-1-neg N/A

          \[\leadsto \left(0 - i\right) \cdot \left(-1 \cdot y + \color{blue}{\left(\mathsf{neg}\left(\frac{a + \left(t + z\right)}{i}\right)\right)}\right) \]

       7. unsub-neg N/A

          \[\leadsto \left(0 - i\right) \cdot \color{blue}{\left(-1 \cdot y - \frac{a + \left(t + z\right)}{i}\right)} \]

       8. --lowering--.f64 N/A

          \[\leadsto \left(0 - i\right) \cdot \color{blue}{\left(-1 \cdot y - \frac{a + \left(t + z\right)}{i}\right)} \]

       9. mul-1-neg N/A

          \[\leadsto \left(0 - i\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)} - \frac{a + \left(t + z\right)}{i}\right) \]

       10. neg-sub0 N/A

           \[\leadsto \left(0 - i\right) \cdot \left(\color{blue}{\left(0 - y\right)} - \frac{a + \left(t + z\right)}{i}\right) \]

       11. --lowering--.f64 N/A

           \[\leadsto \left(0 - i\right) \cdot \left(\color{blue}{\left(0 - y\right)} - \frac{a + \left(t + z\right)}{i}\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \left(0 - i\right) \cdot \left(\left(0 - y\right) - \color{blue}{\frac{a + \left(t + z\right)}{i}}\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \left(0 - i\right) \cdot \left(\left(0 - y\right) - \frac{\color{blue}{a + \left(t + z\right)}}{i}\right) \]

       14. +-lowering-+.f64 79.5

           \[\leadsto \left(0 - i\right) \cdot \left(\left(0 - y\right) - \frac{a + \color{blue}{\left(t + z\right)}}{i}\right) \]

   11. Simplified 79.5%

       \[\leadsto \color{blue}{\left(0 - i\right) \cdot \left(\left(0 - y\right) - \frac{a + \left(t + z\right)}{i}\right)} \]

   if -9.6e62 < i < 2e46

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\color{blue}{z \cdot \left(1 + \frac{x \cdot \log y}{z}\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{z \cdot \left(1 + \frac{x \cdot \log y}{z}\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\left(z \cdot \color{blue}{\left(\frac{x \cdot \log y}{z} + 1\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      3. associate-/l* N/A

         \[\leadsto \left(\left(\left(z \cdot \left(\color{blue}{x \cdot \frac{\log y}{z}} + 1\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\left(z \cdot \color{blue}{\mathsf{fma}\left(x, \frac{\log y}{z}, 1\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(z \cdot \mathsf{fma}\left(x, \color{blue}{\frac{\log y}{z}}, 1\right) + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      6. log-lowering-log.f64 88.6

         \[\leadsto \left(\left(\left(z \cdot \mathsf{fma}\left(x, \frac{\color{blue}{\log y}}{z}, 1\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   5. Simplified 88.6%

      \[\leadsto \left(\left(\left(\color{blue}{z \cdot \mathsf{fma}\left(x, \frac{\log y}{z}, 1\right)} + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right) + a} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{t + \left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{t + \left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto t + \color{blue}{\left(\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} \]

      5. +-commutative N/A

         \[\leadsto t + \left(\color{blue}{\left(\left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right) + z\right)} + a\right) \]

      6. associate-+l+ N/A

         \[\leadsto t + \left(\color{blue}{\left(i \cdot y + \left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)\right)} + a\right) \]

      7. *-commutative N/A

         \[\leadsto t + \left(\left(\color{blue}{y \cdot i} + \left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)\right) + a\right) \]

      8. +-commutative N/A

         \[\leadsto t + \left(\left(y \cdot i + \color{blue}{\left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)}\right) + a\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto t + \left(\color{blue}{\mathsf{fma}\left(y, i, z + \log c \cdot \left(b - \frac{1}{2}\right)\right)} + a\right) \]

      10. +-commutative N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + z}\right) + a\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z\right)}\right) + a\right) \]

      12. log-lowering-log.f64 N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z\right)\right) + a\right) \]

      13. sub-neg N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z\right)\right) + a\right) \]

      14. metadata-eval N/A

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z\right)\right) + a\right) \]

      15. +-lowering-+.f64 83.5

          \[\leadsto t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + -0.5}, z\right)\right) + a\right) \]

   8. Simplified 83.5%

      \[\leadsto \color{blue}{t + \left(\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right) + a\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto t + \left(\color{blue}{\left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)} + a\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto t + \left(\color{blue}{\left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)} + a\right) \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto t + \left(\color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z\right)} + a\right) \]

       3. log-lowering-log.f64 N/A

          \[\leadsto t + \left(\mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z\right) + a\right) \]

       4. sub-neg N/A

          \[\leadsto t + \left(\mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z\right) + a\right) \]

       5. metadata-eval N/A

          \[\leadsto t + \left(\mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z\right) + a\right) \]

       6. +-lowering-+.f64 81.1

          \[\leadsto t + \left(\mathsf{fma}\left(\log c, \color{blue}{b + -0.5}, z\right) + a\right) \]

   11. Simplified 81.1%

       \[\leadsto t + \left(\color{blue}{\mathsf{fma}\left(\log c, b + -0.5, z\right)} + a\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -9.6 \cdot 10^{+62}:\\ \;\;\;\;i \cdot \left(y + \frac{a + \left(z + t\right)}{i}\right)\\ \mathbf{elif}\;i \leq 2 \cdot 10^{+46}:\\ \;\;\;\;t + \left(a + \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(y + \frac{a + \left(z + t\right)}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 57.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.12 \cdot 10^{-159}:\\ \;\;\;\;t + \mathsf{fma}\left(y, i, z\right)\\ \mathbf{elif}\;a \leq 7.8 \cdot 10^{-115}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t + \mathsf{fma}\left(i, y, z\right)}{a}, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= a 1.12e-159)
   (+ t (fma y i z))
   (if (<= a 7.8e-115) (* x (log y)) (fma a (/ (+ t (fma i y z)) a) a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (a <= 1.12e-159) {
		tmp = t + fma(y, i, z);
	} else if (a <= 7.8e-115) {
		tmp = x * log(y);
	} else {
		tmp = fma(a, ((t + fma(i, y, z)) / a), a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (a <= 1.12e-159)
		tmp = Float64(t + fma(y, i, z));
	elseif (a <= 7.8e-115)
		tmp = Float64(x * log(y));
	else
		tmp = fma(a, Float64(Float64(t + fma(i, y, z)) / a), a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[a, 1.12e-159], N[(t + N[(y * i + z), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 7.8e-115], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(t + N[(i * y + z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.12000000000000006e-159

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)} \]

      2. sub-neg N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{-1}\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right) \cdot \left(-1 \cdot a\right) + -1 \cdot \left(-1 \cdot a\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right)} + -1 \cdot \left(-1 \cdot a\right) \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot a\right) \cdot -1\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}} + -1 \cdot \left(-1 \cdot a\right) \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(-1 \cdot a\right)\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot -1\right) \cdot a\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\color{blue}{1} \cdot a\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      10. *-lft-identity N/A

          \[\leadsto \color{blue}{a} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      11. associate-*r* N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{\left(-1 \cdot -1\right) \cdot a} \]

      12. metadata-eval N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{1} \cdot a \]

      13. *-lft-identity N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{a} \]

   5. Simplified 68.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, t\right)\right)}{a}, a\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \color{blue}{t}}{a}, a\right) \]
   7. Step-by-step derivation

   8. Simplified 47.4%

      \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \color{blue}{t}}{a}, a\right) \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{t + \left(z + i \cdot y\right)} \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{t + \left(z + i \cdot y\right)} \]

       2. +-commutative N/A

          \[\leadsto t + \color{blue}{\left(i \cdot y + z\right)} \]

       3. *-commutative N/A

          \[\leadsto t + \left(\color{blue}{y \cdot i} + z\right) \]

       4. accelerator-lowering-fma.f64 51.1

          \[\leadsto t + \color{blue}{\mathsf{fma}\left(y, i, z\right)} \]

   11. Simplified 51.1%

       \[\leadsto \color{blue}{t + \mathsf{fma}\left(y, i, z\right)} \]

   if 1.12000000000000006e-159 < a < 7.7999999999999997e-115

   1. Initial program 99.6%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \log y} \]

      2. log-lowering-log.f64 44.0

         \[\leadsto x \cdot \color{blue}{\log y} \]

   5. Simplified 44.0%

      \[\leadsto \color{blue}{x \cdot \log y} \]

   if 7.7999999999999997e-115 < a

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)} \]

      2. sub-neg N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{-1}\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right) \cdot \left(-1 \cdot a\right) + -1 \cdot \left(-1 \cdot a\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right)} + -1 \cdot \left(-1 \cdot a\right) \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot a\right) \cdot -1\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}} + -1 \cdot \left(-1 \cdot a\right) \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(-1 \cdot a\right)\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot -1\right) \cdot a\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\color{blue}{1} \cdot a\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      10. *-lft-identity N/A

          \[\leadsto \color{blue}{a} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      11. associate-*r* N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{\left(-1 \cdot -1\right) \cdot a} \]

      12. metadata-eval N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{1} \cdot a \]

      13. *-lft-identity N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{a} \]

   5. Simplified 90.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, t\right)\right)}{a}, a\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \color{blue}{t}}{a}, a\right) \]
   7. Step-by-step derivation

   8. Simplified 70.7%

      \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \color{blue}{t}}{a}, a\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.12 \cdot 10^{-159}:\\ \;\;\;\;t + \mathsf{fma}\left(y, i, z\right)\\ \mathbf{elif}\;a \leq 7.8 \cdot 10^{-115}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t + \mathsf{fma}\left(i, y, z\right)}{a}, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 60.2% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 32:\\ \;\;\;\;t + \mathsf{fma}\left(y, i, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t + \mathsf{fma}\left(i, y, z\right)}{a}, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= a 32.0) (+ t (fma y i z)) (fma a (/ (+ t (fma i y z)) a) a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (a <= 32.0) {
		tmp = t + fma(y, i, z);
	} else {
		tmp = fma(a, ((t + fma(i, y, z)) / a), a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (a <= 32.0)
		tmp = Float64(t + fma(y, i, z));
	else
		tmp = fma(a, Float64(Float64(t + fma(i, y, z)) / a), a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[a, 32.0], N[(t + N[(y * i + z), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(t + N[(i * y + z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 32

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)} \]

      2. sub-neg N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{-1}\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right) \cdot \left(-1 \cdot a\right) + -1 \cdot \left(-1 \cdot a\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right)} + -1 \cdot \left(-1 \cdot a\right) \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot a\right) \cdot -1\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}} + -1 \cdot \left(-1 \cdot a\right) \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(-1 \cdot a\right)\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot -1\right) \cdot a\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\color{blue}{1} \cdot a\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      10. *-lft-identity N/A

          \[\leadsto \color{blue}{a} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      11. associate-*r* N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{\left(-1 \cdot -1\right) \cdot a} \]

      12. metadata-eval N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{1} \cdot a \]

      13. *-lft-identity N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{a} \]

   5. Simplified 67.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, t\right)\right)}{a}, a\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \color{blue}{t}}{a}, a\right) \]
   7. Step-by-step derivation

   8. Simplified 47.9%

      \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \color{blue}{t}}{a}, a\right) \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{t + \left(z + i \cdot y\right)} \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{t + \left(z + i \cdot y\right)} \]

       2. +-commutative N/A

          \[\leadsto t + \color{blue}{\left(i \cdot y + z\right)} \]

       3. *-commutative N/A

          \[\leadsto t + \left(\color{blue}{y \cdot i} + z\right) \]

       4. accelerator-lowering-fma.f64 54.2

          \[\leadsto t + \color{blue}{\mathsf{fma}\left(y, i, z\right)} \]

   11. Simplified 54.2%

       \[\leadsto \color{blue}{t + \mathsf{fma}\left(y, i, z\right)} \]

   if 32 < a

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} - 1\right)} \]

      2. sub-neg N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{-1}\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right) \cdot \left(-1 \cdot a\right) + -1 \cdot \left(-1 \cdot a\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(-1 \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}\right)} + -1 \cdot \left(-1 \cdot a\right) \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot a\right) \cdot -1\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a}} + -1 \cdot \left(-1 \cdot a\right) \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(-1 \cdot a\right)\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot -1\right) \cdot a\right)} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\color{blue}{1} \cdot a\right) \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      10. *-lft-identity N/A

          \[\leadsto \color{blue}{a} \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + -1 \cdot \left(-1 \cdot a\right) \]

      11. associate-*r* N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{\left(-1 \cdot -1\right) \cdot a} \]

      12. metadata-eval N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{1} \cdot a \]

      13. *-lft-identity N/A

          \[\leadsto a \cdot \frac{t + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{a} + \color{blue}{a} \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, t\right)\right)}{a}, a\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \color{blue}{t}}{a}, a\right) \]
   7. Step-by-step derivation

   8. Simplified 78.5%

      \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(i, y, z\right) + \color{blue}{t}}{a}, a\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 32:\\ \;\;\;\;t + \mathsf{fma}\left(y, i, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t + \mathsf{fma}\left(i, y, z\right)}{a}, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 17.0% accurate, 234.0× speedup

Math

\[\begin{array}{l} \\ a \end{array} \]

FPCore

(FPCore (x y z t a b c i) :precision binary64 a)

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return a;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = a
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return a;
}

Python

def code(x, y, z, t, a, b, c, i):
	return a

Julia

function code(x, y, z, t, a, b, c, i)
	return a
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = a;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := a

Derivation

1. Initial program 99.9%

   \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
2. Add Preprocessing

3. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a} \]
4. Step-by-step derivation

5. Simplified 15.7%

   \[\leadsto \color{blue}{a} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024195 
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, B"
  :precision binary64
  (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))