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

Percentage Accurate: 99.9% → 99.9%

Time: 15.1s

Alternatives: 17

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: 99.9% 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: 99.9% 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 99.8%

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

Alternative 2: 40.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \left(\left(\left(\left(t\_1 + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(i, y, t\_1\right)\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, z\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+226}:\\ \;\;\;\;a + b \cdot \log c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

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))))
   (if (<= t_2 -1e+300)
     (fma i y t_1)
     (if (<= t_2 -4e+82)
       (fma x (log y) z)
       (if (<= t_2 2e+226) (+ a (* b (log c))) (fma y i 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);
	double t_2 = ((((t_1 + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_2 <= -1e+300) {
		tmp = fma(i, y, t_1);
	} else if (t_2 <= -4e+82) {
		tmp = fma(x, log(y), z);
	} else if (t_2 <= 2e+226) {
		tmp = a + (b * log(c));
	} else {
		tmp = fma(y, i, a);
	}
	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))
	tmp = 0.0
	if (t_2 <= -1e+300)
		tmp = fma(i, y, t_1);
	elseif (t_2 <= -4e+82)
		tmp = fma(x, log(y), z);
	elseif (t_2 <= 2e+226)
		tmp = Float64(a + Float64(b * log(c)));
	else
		tmp = fma(y, i, a);
	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]}, If[LessEqual[t$95$2, -1e+300], N[(i * y + t$95$1), $MachinePrecision], If[LessEqual[t$95$2, -4e+82], N[(x * N[Log[y], $MachinePrecision] + z), $MachinePrecision], If[LessEqual[t$95$2, 2e+226], N[(a + N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * i + a), $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)) < -1.0000000000000001e300

   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 t around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-log.f64 99.9

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

   5. Simplified 99.9%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 99.9

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. lower-log.f64 86.5

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

   11. Simplified 86.5%

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

   if -1.0000000000000001e300 < (+.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)) < -3.9999999999999999e82

   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

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \log y + -1 \cdot \frac{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{x}\right)\right)} \]
   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. lower-*.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. lower-+.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. lower-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. lower-/.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 62.1%

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

   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. lower-/.f64 31.8

         \[\leadsto x \cdot \left(\log y + \color{blue}{\frac{z}{x}}\right) \]

   8. Simplified 31.8%

      \[\leadsto x \cdot \left(\log y + \color{blue}{\frac{z}{x}}\right) \]

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-log.f64 38.7

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

   11. Simplified 38.7%

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

   if -3.9999999999999999e82 < (+.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)) < 1.99999999999999992e226

   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 t around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-log.f64 82.7

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

   5. Simplified 82.7%

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

   6. Taylor expanded in b around inf

      \[\leadsto a + \color{blue}{b \cdot \log c} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto a + \color{blue}{\log c \cdot b} \]

      2. lower-*.f64 N/A

         \[\leadsto a + \color{blue}{\log c \cdot b} \]

      3. lower-log.f64 41.9

         \[\leadsto a + \color{blue}{\log c} \cdot b \]

   8. Simplified 41.9%

      \[\leadsto a + \color{blue}{\log c \cdot b} \]

   if 1.99999999999999992e226 < (+.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 t around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-log.f64 87.2

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

   5. Simplified 87.2%

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

   6. Taylor expanded in i around inf

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

      1. lower-*.f64 56.1

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

   8. Simplified 56.1%

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

   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. *-commutative N/A

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

       3. lower-fma.f64 56.1

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

   11. Simplified 56.1%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -1 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(i, y, x \cdot \log y\right)\\ \mathbf{elif}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -4 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, z\right)\\ \mathbf{elif}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq 2 \cdot 10^{+226}:\\ \;\;\;\;a + b \cdot \log c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 40.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+303}:\\ \;\;\;\;y \cdot \left(i + \frac{z}{y}\right)\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, z\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+226}:\\ \;\;\;\;a + b \cdot \log c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

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 -5e+303)
     (* y (+ i (/ z y)))
     (if (<= t_1 -4e+82)
       (fma x (log y) z)
       (if (<= t_1 2e+226) (+ a (* b (log c))) (fma y i 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 <= -5e+303) {
		tmp = y * (i + (z / y));
	} else if (t_1 <= -4e+82) {
		tmp = fma(x, log(y), z);
	} else if (t_1 <= 2e+226) {
		tmp = a + (b * log(c));
	} else {
		tmp = fma(y, i, 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 <= -5e+303)
		tmp = Float64(y * Float64(i + Float64(z / y)));
	elseif (t_1 <= -4e+82)
		tmp = fma(x, log(y), z);
	elseif (t_1 <= 2e+226)
		tmp = Float64(a + Float64(b * log(c)));
	else
		tmp = fma(y, i, 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, -5e+303], N[(y * N[(i + N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -4e+82], N[(x * N[Log[y], $MachinePrecision] + z), $MachinePrecision], If[LessEqual[t$95$1, 2e+226], N[(a + N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * i + a), $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)) < -4.9999999999999997e303

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 N/A

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

   5. Simplified 84.0%

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

   6. Taylor expanded in z around inf

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

      1. lower-/.f64 83.6

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

   8. Simplified 83.6%

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

   9. Taylor expanded in y around inf

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-/.f64 89.6

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

   11. Simplified 89.6%

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

   if -4.9999999999999997e303 < (+.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)) < -3.9999999999999999e82

   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

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \log y + -1 \cdot \frac{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{x}\right)\right)} \]
   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. lower-*.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. lower-+.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. lower-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. lower-/.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 62.8%

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

   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. lower-/.f64 32.6

         \[\leadsto x \cdot \left(\log y + \color{blue}{\frac{z}{x}}\right) \]

   8. Simplified 32.6%

      \[\leadsto x \cdot \left(\log y + \color{blue}{\frac{z}{x}}\right) \]

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-log.f64 39.2

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

   11. Simplified 39.2%

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

   if -3.9999999999999999e82 < (+.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)) < 1.99999999999999992e226

   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 t around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-log.f64 82.7

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

   5. Simplified 82.7%

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

   6. Taylor expanded in b around inf

      \[\leadsto a + \color{blue}{b \cdot \log c} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto a + \color{blue}{\log c \cdot b} \]

      2. lower-*.f64 N/A

         \[\leadsto a + \color{blue}{\log c \cdot b} \]

      3. lower-log.f64 41.9

         \[\leadsto a + \color{blue}{\log c} \cdot b \]

   8. Simplified 41.9%

      \[\leadsto a + \color{blue}{\log c \cdot b} \]

   if 1.99999999999999992e226 < (+.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 t around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-log.f64 87.2

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

   5. Simplified 87.2%

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

   6. Taylor expanded in i around inf

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

      1. lower-*.f64 56.1

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

   8. Simplified 56.1%

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

   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. *-commutative N/A

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

       3. lower-fma.f64 56.1

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

   11. Simplified 56.1%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -5 \cdot 10^{+303}:\\ \;\;\;\;y \cdot \left(i + \frac{z}{y}\right)\\ \mathbf{elif}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -4 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, z\right)\\ \mathbf{elif}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq 2 \cdot 10^{+226}:\\ \;\;\;\;a + b \cdot \log c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 39.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+303}:\\ \;\;\;\;y \cdot \left(i + \frac{z}{y}\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

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 -5e+303)
     (* y (+ i (/ z y)))
     (if (<= t_1 5e+85) (fma x (log y) z) (fma y i 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 <= -5e+303) {
		tmp = y * (i + (z / y));
	} else if (t_1 <= 5e+85) {
		tmp = fma(x, log(y), z);
	} else {
		tmp = fma(y, i, 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 <= -5e+303)
		tmp = Float64(y * Float64(i + Float64(z / y)));
	elseif (t_1 <= 5e+85)
		tmp = fma(x, log(y), z);
	else
		tmp = fma(y, i, 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, -5e+303], N[(y * N[(i + N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+85], N[(x * N[Log[y], $MachinePrecision] + z), $MachinePrecision], N[(y * i + 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)) < -4.9999999999999997e303

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 N/A

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

   5. Simplified 84.0%

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

   6. Taylor expanded in z around inf

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

      1. lower-/.f64 83.6

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

   8. Simplified 83.6%

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

   9. Taylor expanded in y around inf

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-/.f64 89.6

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

   11. Simplified 89.6%

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

   if -4.9999999999999997e303 < (+.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)) < 5.0000000000000001e85

   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

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \log y + -1 \cdot \frac{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{x}\right)\right)} \]
   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. lower-*.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. lower-+.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. lower-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. lower-/.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 66.1%

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

   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. lower-/.f64 30.3

         \[\leadsto x \cdot \left(\log y + \color{blue}{\frac{z}{x}}\right) \]

   8. Simplified 30.3%

      \[\leadsto x \cdot \left(\log y + \color{blue}{\frac{z}{x}}\right) \]

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-log.f64 36.2

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

   11. Simplified 36.2%

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

   if 5.0000000000000001e85 < (+.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 99.8%

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

   3. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-log.f64 85.3

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

   5. Simplified 85.3%

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

   6. Taylor expanded in i around inf

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

      1. lower-*.f64 45.1

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

   8. Simplified 45.1%

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

   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. *-commutative N/A

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

       3. lower-fma.f64 45.1

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

   11. Simplified 45.1%

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

Alternative 5: 38.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -1 \cdot 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

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))
      -1e+28)
   (fma y i z)
   (fma y i 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)) <= -1e+28) {
		tmp = fma(y, i, z);
	} else {
		tmp = fma(y, i, 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)) <= -1e+28)
		tmp = fma(y, i, z);
	else
		tmp = fma(y, i, 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], -1e+28], N[(y * i + z), $MachinePrecision], N[(y * i + 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)) < -9.99999999999999958e27

   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

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

      1. lower-*.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 N/A

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

   5. Simplified 68.8%

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

   6. Taylor expanded in z around inf

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

      1. lower-/.f64 30.7

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

   8. Simplified 30.7%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 38.9

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

   11. Simplified 38.9%

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

   if -9.99999999999999958e27 < (+.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 99.8%

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

   3. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-log.f64 85.5

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

   5. Simplified 85.5%

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

   6. Taylor expanded in i around inf

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

      1. lower-*.f64 42.4

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

   8. Simplified 42.4%

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

   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. *-commutative N/A

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

       3. lower-fma.f64 42.4

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

   11. Simplified 42.4%

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

Alternative 6: 92.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(i, y, \mathsf{fma}\left(x, \log y, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\right)\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{+118}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+189}:\\ \;\;\;\;a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma i y (fma x (log y) (fma (log c) (+ b -0.5) z)))))
   (if (<= x -2.6e+118)
     t_1
     (if (<= x 5.5e+189)
       (+ a (+ (fma i y z) (fma (log c) (+ b -0.5) t)))
       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(i, y, fma(x, log(y), fma(log(c), (b + -0.5), z)));
	double tmp;
	if (x <= -2.6e+118) {
		tmp = t_1;
	} else if (x <= 5.5e+189) {
		tmp = a + (fma(i, y, z) + fma(log(c), (b + -0.5), t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(i, y, fma(x, log(y), fma(log(c), Float64(b + -0.5), z)))
	tmp = 0.0
	if (x <= -2.6e+118)
		tmp = t_1;
	elseif (x <= 5.5e+189)
		tmp = Float64(a + Float64(fma(i, y, z) + fma(log(c), Float64(b + -0.5), t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * y + N[(x * N[Log[y], $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.6e+118], t$95$1, If[LessEqual[x, 5.5e+189], N[(a + N[(N[(i * y + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.60000000000000016e118 or 5.5e189 < 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 t around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-log.f64 92.9

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

   5. Simplified 92.9%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 89.9

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

   8. Simplified 89.9%

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

   if -2.60000000000000016e118 < x < 5.5e189

   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. lower-+.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. lower-+.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. lower-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. lower-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. lower-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. lower-+.f64 97.7

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

   5. Simplified 97.7%

      \[\leadsto \color{blue}{a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(i, y, \mathsf{fma}\left(x, \log y, b \cdot \log c\right)\right)\\ \mathbf{if}\;x \leq -2.9 \cdot 10^{+118}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+191}:\\ \;\;\;\;a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma i y (fma x (log y) (* b (log c))))))
   (if (<= x -2.9e+118)
     t_1
     (if (<= x 5.6e+191)
       (+ a (+ (fma i y z) (fma (log c) (+ b -0.5) t)))
       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(i, y, fma(x, log(y), (b * log(c))));
	double tmp;
	if (x <= -2.9e+118) {
		tmp = t_1;
	} else if (x <= 5.6e+191) {
		tmp = a + (fma(i, y, z) + fma(log(c), (b + -0.5), t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(i, y, fma(x, log(y), Float64(b * log(c))))
	tmp = 0.0
	if (x <= -2.9e+118)
		tmp = t_1;
	elseif (x <= 5.6e+191)
		tmp = Float64(a + Float64(fma(i, y, z) + fma(log(c), Float64(b + -0.5), t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * y + N[(x * N[Log[y], $MachinePrecision] + N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.9e+118], t$95$1, If[LessEqual[x, 5.6e+191], N[(a + N[(N[(i * y + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.90000000000000016e118 or 5.5999999999999998e191 < 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 t around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-log.f64 92.9

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

   5. Simplified 92.9%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 89.9

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

   8. Simplified 89.9%

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

   9. Taylor expanded in b around inf

      \[\leadsto \mathsf{fma}\left(i, y, \mathsf{fma}\left(x, \log y, \color{blue}{b \cdot \log c}\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(i, y, \mathsf{fma}\left(x, \log y, \color{blue}{\log c \cdot b}\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, y, \mathsf{fma}\left(x, \log y, \color{blue}{\log c \cdot b}\right)\right) \]

       3. lower-log.f64 86.7

          \[\leadsto \mathsf{fma}\left(i, y, \mathsf{fma}\left(x, \log y, \color{blue}{\log c} \cdot b\right)\right) \]

   11. Simplified 86.7%

       \[\leadsto \mathsf{fma}\left(i, y, \mathsf{fma}\left(x, \log y, \color{blue}{\log c \cdot b}\right)\right) \]

   if -2.90000000000000016e118 < x < 5.5999999999999998e191

   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. lower-+.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. lower-+.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. lower-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. lower-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. lower-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. lower-+.f64 97.7

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

   5. Simplified 97.7%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(i, y, \mathsf{fma}\left(x, \log y, b \cdot \log c\right)\right)\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+191}:\\ \;\;\;\;a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, \mathsf{fma}\left(x, \log y, b \cdot \log c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 t around 0

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

   1. lower-+.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. associate-+r+ N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-log.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-log.f64 85.3

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

5. Simplified 85.3%

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

Alternative 9: 88.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(i, y, x \cdot \log y\right)\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+191}:\\ \;\;\;\;a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log y + \frac{y \cdot i}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x -2.9e+118)
   (fma i y (* x (log y)))
   (if (<= x 5.6e+191)
     (+ a (+ (fma i y z) (fma (log c) (+ b -0.5) t)))
     (* x (+ (log y) (/ (* y i) x))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (x <= -2.9e+118) {
		tmp = fma(i, y, (x * log(y)));
	} else if (x <= 5.6e+191) {
		tmp = a + (fma(i, y, z) + fma(log(c), (b + -0.5), t));
	} else {
		tmp = x * (log(y) + ((y * i) / x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= -2.9e+118)
		tmp = fma(i, y, Float64(x * log(y)));
	elseif (x <= 5.6e+191)
		tmp = Float64(a + Float64(fma(i, y, z) + fma(log(c), Float64(b + -0.5), t)));
	else
		tmp = Float64(x * Float64(log(y) + Float64(Float64(y * i) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, -2.9e+118], N[(i * y + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.6e+191], 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[(N[(y * i), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.90000000000000016e118

   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 t around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-log.f64 90.9

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

   5. Simplified 90.9%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 86.0

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

   8. Simplified 86.0%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. lower-log.f64 75.1

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

   11. Simplified 75.1%

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

   if -2.90000000000000016e118 < x < 5.5999999999999998e191

   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. lower-+.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. lower-+.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. lower-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. lower-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. lower-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. lower-+.f64 97.7

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

   5. Simplified 97.7%

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

   if 5.5999999999999998e191 < x

   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

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \log y + -1 \cdot \frac{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)}{x}\right)\right)} \]
   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. lower-*.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. lower-+.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. lower-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. lower-/.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 99.4%

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

   6. Taylor expanded in i around inf

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

      1. lower-*.f64 92.4

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

   8. Simplified 92.4%

      \[\leadsto x \cdot \left(\log y + \frac{\color{blue}{i \cdot y}}{x}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(i, y, x \cdot \log y\right)\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+191}:\\ \;\;\;\;a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log y + \frac{y \cdot i}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(i, y, x \cdot \log y\right)\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{+118}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(i, a \cdot \frac{y}{a}, a\right)\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+191}:\\ \;\;\;\;a + \left(t + \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma i y (* x (log y)))))
   (if (<= x -2.6e+118)
     t_1
     (if (<= x -6.2e-61)
       (fma i (* a (/ y a)) a)
       (if (<= x 2.1e+191) (+ a (+ t (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 = fma(i, y, (x * log(y)));
	double tmp;
	if (x <= -2.6e+118) {
		tmp = t_1;
	} else if (x <= -6.2e-61) {
		tmp = fma(i, (a * (y / a)), a);
	} else if (x <= 2.1e+191) {
		tmp = a + (t + 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 = fma(i, y, Float64(x * log(y)))
	tmp = 0.0
	if (x <= -2.6e+118)
		tmp = t_1;
	elseif (x <= -6.2e-61)
		tmp = fma(i, Float64(a * Float64(y / a)), a);
	elseif (x <= 2.1e+191)
		tmp = Float64(a + Float64(t + 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 * y + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.6e+118], t$95$1, If[LessEqual[x, -6.2e-61], N[(i * N[(a * N[(y / a), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision], If[LessEqual[x, 2.1e+191], N[(a + N[(t + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.60000000000000016e118 or 2.1000000000000001e191 < 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 t around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-log.f64 92.9

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

   5. Simplified 92.9%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 89.9

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

   8. Simplified 89.9%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. lower-log.f64 81.9

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

   11. Simplified 81.9%

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

   if -2.60000000000000016e118 < x < -6.1999999999999999e-61

   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 t around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-log.f64 84.5

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

   5. Simplified 84.5%

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

   6. Taylor expanded in i around inf

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

      1. lower-*.f64 65.2

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

   8. Simplified 65.2%

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

   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. *-commutative N/A

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

       3. lower-fma.f64 65.2

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

   11. Simplified 65.2%

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

   12. Taylor expanded in a around inf

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\frac{i \cdot y}{a} \cdot a + 1 \cdot a} \]

       3. *-lft-identity N/A

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

       4. associate-/l* N/A

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

       5. associate-*l* N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 62.8

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

   14. Simplified 62.8%

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

   if -6.1999999999999999e-61 < x < 2.1000000000000001e191

   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. lower-+.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. lower-+.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. lower-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. lower-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. lower-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. lower-+.f64 97.9

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

   5. Simplified 97.9%

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

   6. Taylor expanded in i around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 75.0

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

   8. Simplified 75.0%

      \[\leadsto \color{blue}{a + \left(\mathsf{fma}\left(\log c, b + -0.5, z\right) + t\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(i, y, x \cdot \log y\right)\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(i, a \cdot \frac{y}{a}, a\right)\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+191}:\\ \;\;\;\;a + \left(t + \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, x \cdot \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 88.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(i, y, x \cdot \log y\right)\\ \mathbf{if}\;x \leq -2.9 \cdot 10^{+118}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+191}:\\ \;\;\;\;a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma i y (* x (log y)))))
   (if (<= x -2.9e+118)
     t_1
     (if (<= x 5.6e+191)
       (+ a (+ (fma i y z) (fma (log c) (+ b -0.5) t)))
       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(i, y, (x * log(y)));
	double tmp;
	if (x <= -2.9e+118) {
		tmp = t_1;
	} else if (x <= 5.6e+191) {
		tmp = a + (fma(i, y, z) + fma(log(c), (b + -0.5), t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(i, y, Float64(x * log(y)))
	tmp = 0.0
	if (x <= -2.9e+118)
		tmp = t_1;
	elseif (x <= 5.6e+191)
		tmp = Float64(a + Float64(fma(i, y, z) + fma(log(c), Float64(b + -0.5), t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * y + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.9e+118], t$95$1, If[LessEqual[x, 5.6e+191], N[(a + N[(N[(i * y + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.90000000000000016e118 or 5.5999999999999998e191 < 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 t around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-log.f64 92.9

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

   5. Simplified 92.9%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 89.9

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

   8. Simplified 89.9%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. lower-log.f64 81.9

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

   11. Simplified 81.9%

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

   if -2.90000000000000016e118 < x < 5.5999999999999998e191

   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. lower-+.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. lower-+.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. lower-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. lower-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. lower-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. lower-+.f64 97.7

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

   5. Simplified 97.7%

      \[\leadsto \color{blue}{a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 53.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(i, y, x \cdot \log y\right)\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{+118}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(i, a \cdot \frac{y}{a}, a\right)\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+191}:\\ \;\;\;\;a + \mathsf{fma}\left(\log c, b + -0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma i y (* x (log y)))))
   (if (<= x -2.6e+118)
     t_1
     (if (<= x -6.2e-61)
       (fma i (* a (/ y a)) a)
       (if (<= x 2.1e+191) (+ 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 = fma(i, y, (x * log(y)));
	double tmp;
	if (x <= -2.6e+118) {
		tmp = t_1;
	} else if (x <= -6.2e-61) {
		tmp = fma(i, (a * (y / a)), a);
	} else if (x <= 2.1e+191) {
		tmp = 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 = fma(i, y, Float64(x * log(y)))
	tmp = 0.0
	if (x <= -2.6e+118)
		tmp = t_1;
	elseif (x <= -6.2e-61)
		tmp = fma(i, Float64(a * Float64(y / a)), a);
	elseif (x <= 2.1e+191)
		tmp = 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 * y + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.6e+118], t$95$1, If[LessEqual[x, -6.2e-61], N[(i * N[(a * N[(y / a), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision], If[LessEqual[x, 2.1e+191], N[(a + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.60000000000000016e118 or 2.1000000000000001e191 < 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 t around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-log.f64 92.9

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

   5. Simplified 92.9%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 89.9

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

   8. Simplified 89.9%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. lower-log.f64 81.9

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

   11. Simplified 81.9%

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

   if -2.60000000000000016e118 < x < -6.1999999999999999e-61

   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 t around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-log.f64 84.5

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

   5. Simplified 84.5%

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

   6. Taylor expanded in i around inf

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

      1. lower-*.f64 65.2

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

   8. Simplified 65.2%

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

   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. *-commutative N/A

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

       3. lower-fma.f64 65.2

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

   11. Simplified 65.2%

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

   12. Taylor expanded in a around inf

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\frac{i \cdot y}{a} \cdot a + 1 \cdot a} \]

       3. *-lft-identity N/A

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

       4. associate-/l* N/A

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

       5. associate-*l* N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 62.8

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

   14. Simplified 62.8%

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

   if -6.1999999999999999e-61 < x < 2.1000000000000001e191

   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 t around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-log.f64 82.5

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

   5. Simplified 82.5%

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

   6. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-log.f64 78.8

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

   8. Simplified 78.8%

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

   9. Taylor expanded in i around 0

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

       1. +-commutative N/A

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

       2. associate-+l+ N/A

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

       3. mul-1-neg N/A

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

       4. distribute-rgt-neg-in N/A

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

       5. sub-neg N/A

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

       6. mul-1-neg N/A

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

       7. distribute-neg-in N/A

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

       8. associate-/l* N/A

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

       9. rgt-mult-inverse N/A

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

       10. distribute-lft-in N/A

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

       11. +-commutative N/A

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

       12. remove-double-neg N/A

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

   11. Simplified 57.5%

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

   12. Taylor expanded in x around 0

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

       1. lower-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-log.f64 N/A

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. lower-+.f64 58.1

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

   14. Simplified 58.1%

       \[\leadsto \color{blue}{a + \mathsf{fma}\left(\log c, b + -0.5, z\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(i, y, x \cdot \log y\right)\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(i, a \cdot \frac{y}{a}, a\right)\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+191}:\\ \;\;\;\;a + \mathsf{fma}\left(\log c, b + -0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, x \cdot \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(i, y, x \cdot \log y\right)\\ \mathbf{if}\;x \leq -2.9 \cdot 10^{+118}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+191}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right) + \mathsf{fma}\left(\log c, b + -0.5, a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma i y (* x (log y)))))
   (if (<= x -2.9e+118)
     t_1
     (if (<= x 5.6e+191) (+ (fma y i z) (fma (log c) (+ b -0.5) 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 = fma(i, y, (x * log(y)));
	double tmp;
	if (x <= -2.9e+118) {
		tmp = t_1;
	} else if (x <= 5.6e+191) {
		tmp = fma(y, i, z) + fma(log(c), (b + -0.5), a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(i, y, Float64(x * log(y)))
	tmp = 0.0
	if (x <= -2.9e+118)
		tmp = t_1;
	elseif (x <= 5.6e+191)
		tmp = Float64(fma(y, i, z) + fma(log(c), Float64(b + -0.5), a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * y + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.9e+118], t$95$1, If[LessEqual[x, 5.6e+191], N[(N[(y * i + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.90000000000000016e118 or 5.5999999999999998e191 < 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 t around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-log.f64 92.9

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

   5. Simplified 92.9%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 89.9

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

   8. Simplified 89.9%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. lower-log.f64 81.9

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

   11. Simplified 81.9%

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

   if -2.90000000000000016e118 < x < 5.5999999999999998e191

   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 t around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-log.f64 82.8

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

   5. Simplified 82.8%

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

   6. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-log.f64 78.2

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

   8. Simplified 78.2%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate-+r+ N/A

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

       3. associate-+l+ N/A

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

       4. lower-+.f64 N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-log.f64 N/A

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

       10. sub-neg N/A

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

       11. metadata-eval N/A

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

       12. lower-+.f64 80.6

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

   11. Simplified 80.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, z\right) + \mathsf{fma}\left(\log c, b + -0.5, a\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 72.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(i, y, x \cdot \log y\right)\\ \mathbf{if}\;x \leq -2.9 \cdot 10^{+118}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+191}:\\ \;\;\;\;a + \left(\mathsf{fma}\left(i, y, z\right) + b \cdot \log c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma i y (* x (log y)))))
   (if (<= x -2.9e+118)
     t_1
     (if (<= x 5.6e+191) (+ a (+ (fma i y z) (* b (log c)))) 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(i, y, (x * log(y)));
	double tmp;
	if (x <= -2.9e+118) {
		tmp = t_1;
	} else if (x <= 5.6e+191) {
		tmp = a + (fma(i, y, z) + (b * log(c)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(i, y, Float64(x * log(y)))
	tmp = 0.0
	if (x <= -2.9e+118)
		tmp = t_1;
	elseif (x <= 5.6e+191)
		tmp = Float64(a + Float64(fma(i, y, z) + Float64(b * log(c))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * y + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.9e+118], t$95$1, If[LessEqual[x, 5.6e+191], N[(a + N[(N[(i * y + z), $MachinePrecision] + N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.90000000000000016e118 or 5.5999999999999998e191 < 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 t around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-log.f64 92.9

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

   5. Simplified 92.9%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 89.9

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

   8. Simplified 89.9%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. lower-log.f64 81.9

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

   11. Simplified 81.9%

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

   if -2.90000000000000016e118 < x < 5.5999999999999998e191

   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. lower-+.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. lower-+.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. lower-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. lower-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. lower-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. lower-+.f64 97.7

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

   5. Simplified 97.7%

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

   6. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 78.6

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

   8. Simplified 78.6%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(i, y, x \cdot \log y\right)\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+191}:\\ \;\;\;\;a + \left(\mathsf{fma}\left(i, y, z\right) + b \cdot \log c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, x \cdot \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.6 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y \cdot i}{a}, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= a 1.6e+154)
   (fma y i (fma (log c) (+ b -0.5) z))
   (fma a (/ (* y i) 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 <= 1.6e+154) {
		tmp = fma(y, i, fma(log(c), (b + -0.5), z));
	} else {
		tmp = fma(a, ((y * i) / a), a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (a <= 1.6e+154)
		tmp = fma(y, i, fma(log(c), Float64(b + -0.5), z));
	else
		tmp = fma(a, Float64(Float64(y * i) / a), a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[a, 1.6e+154], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(y * i), $MachinePrecision] / a), $MachinePrecision] + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.6e154

   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 t around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-log.f64 84.8

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

   5. Simplified 84.8%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, y, \mathsf{fma}\left(x, \log y, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z\right)}\right)\right) \]

      12. lower-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, y, \mathsf{fma}\left(x, \log y, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(i, y, \mathsf{fma}\left(x, \log y, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(i, y, \mathsf{fma}\left(x, \log y, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z\right)\right)\right) \]

      15. lower-+.f64 76.7

          \[\leadsto \mathsf{fma}\left(i, y, \mathsf{fma}\left(x, \log y, \mathsf{fma}\left(\log c, \color{blue}{b + -0.5}, z\right)\right)\right) \]

   8. Simplified 76.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, y, \mathsf{fma}\left(x, \log y, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right) + z} \]

       2. associate-+l+ N/A

          \[\leadsto \color{blue}{i \cdot y + \left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{y \cdot i} + \left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right) \]

       4. +-commutative N/A

          \[\leadsto y \cdot i + \color{blue}{\left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)} \]

       5. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, z + \log c \cdot \left(b - \frac{1}{2}\right)\right)} \]

       6. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + z}\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z\right)}\right) \]

       8. lower-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z\right)\right) \]

       9. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z\right)\right) \]

       11. lower-+.f64 57.5

           \[\leadsto \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, \color{blue}{b + -0.5}, z\right)\right) \]

   11. Simplified 57.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b + -0.5, z\right)\right)} \]

   if 1.6e154 < 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 t around 0

      \[\leadsto \color{blue}{a + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \color{blue}{a + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto a + \color{blue}{\left(\left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + z\right)} \]

      3. associate-+l+ N/A

         \[\leadsto a + \color{blue}{\left(i \cdot y + \left(\left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right) + z\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto a + \left(i \cdot y + \color{blue}{\left(z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto a + \color{blue}{\mathsf{fma}\left(i, y, z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      6. associate-+r+ N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\left(z + x \cdot \log y\right) + \log c \cdot \left(b - \frac{1}{2}\right)}\right) \]

      7. +-commutative N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + \left(z + x \cdot \log y\right)}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z + x \cdot \log y\right)}\right) \]

      9. lower-log.f64 N/A

         \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z + x \cdot \log y\right)\right) \]

      10. sub-neg N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z + x \cdot \log y\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z + x \cdot \log y\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, \color{blue}{b + \frac{-1}{2}}, z + x \cdot \log y\right)\right) \]

      13. +-commutative N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \frac{-1}{2}, \color{blue}{x \cdot \log y + z}\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \frac{-1}{2}, \color{blue}{\mathsf{fma}\left(x, \log y, z\right)}\right)\right) \]

      15. lower-log.f64 88.0

          \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \color{blue}{\log y}, z\right)\right)\right) \]

   5. Simplified 88.0%

      \[\leadsto \color{blue}{a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, z\right)\right)\right)} \]

   6. Taylor expanded in i around inf

      \[\leadsto a + \color{blue}{i \cdot y} \]
   7. Step-by-step derivation

      1. lower-*.f64 75.5

         \[\leadsto a + \color{blue}{i \cdot y} \]

   8. Simplified 75.5%

      \[\leadsto a + \color{blue}{i \cdot y} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(1 + \frac{i \cdot y}{a}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto a \cdot \color{blue}{\left(\frac{i \cdot y}{a} + 1\right)} \]

       2. distribute-lft-in N/A

          \[\leadsto \color{blue}{a \cdot \frac{i \cdot y}{a} + a \cdot 1} \]

       3. *-rgt-identity N/A

          \[\leadsto a \cdot \frac{i \cdot y}{a} + \color{blue}{a} \]

       4. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{i \cdot y}{a}, a\right)} \]

       5. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{i \cdot y}{a}}, a\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{y \cdot i}}{a}, a\right) \]

       7. lower-*.f64 75.5

          \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{y \cdot i}}{a}, a\right) \]

   11. Simplified 75.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{y \cdot i}{a}, a\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 38.7% accurate, 33.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, i, a\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i) :precision binary64 (fma y i a))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return fma(y, i, a);
}

Julia

function code(x, y, z, t, a, b, c, i)
	return fma(y, i, a)
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(y * i + a), $MachinePrecision]

Derivation

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 t around 0

   \[\leadsto \color{blue}{a + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
4. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \color{blue}{a + \left(z + \left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto a + \color{blue}{\left(\left(i \cdot y + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + z\right)} \]

   3. associate-+l+ N/A

      \[\leadsto a + \color{blue}{\left(i \cdot y + \left(\left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right) + z\right)\right)} \]

   4. +-commutative N/A

      \[\leadsto a + \left(i \cdot y + \color{blue}{\left(z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)}\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto a + \color{blue}{\mathsf{fma}\left(i, y, z + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

   6. associate-+r+ N/A

      \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\left(z + x \cdot \log y\right) + \log c \cdot \left(b - \frac{1}{2}\right)}\right) \]

   7. +-commutative N/A

      \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\log c \cdot \left(b - \frac{1}{2}\right) + \left(z + x \cdot \log y\right)}\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto a + \mathsf{fma}\left(i, y, \color{blue}{\mathsf{fma}\left(\log c, b - \frac{1}{2}, z + x \cdot \log y\right)}\right) \]

   9. lower-log.f64 N/A

      \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\color{blue}{\log c}, b - \frac{1}{2}, z + x \cdot \log y\right)\right) \]

   10. sub-neg N/A

       \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, \color{blue}{b + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, z + x \cdot \log y\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \color{blue}{\frac{-1}{2}}, z + x \cdot \log y\right)\right) \]

   12. lower-+.f64 N/A

       \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, \color{blue}{b + \frac{-1}{2}}, z + x \cdot \log y\right)\right) \]

   13. +-commutative N/A

       \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \frac{-1}{2}, \color{blue}{x \cdot \log y + z}\right)\right) \]

   14. lower-fma.f64 N/A

       \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + \frac{-1}{2}, \color{blue}{\mathsf{fma}\left(x, \log y, z\right)}\right)\right) \]

   15. lower-log.f64 85.3

       \[\leadsto a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \color{blue}{\log y}, z\right)\right)\right) \]

5. Simplified 85.3%

   \[\leadsto \color{blue}{a + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b + -0.5, \mathsf{fma}\left(x, \log y, z\right)\right)\right)} \]

6. Taylor expanded in i around inf

   \[\leadsto a + \color{blue}{i \cdot y} \]
7. Step-by-step derivation

   1. lower-*.f64 40.9

      \[\leadsto a + \color{blue}{i \cdot y} \]

8. Simplified 40.9%

   \[\leadsto a + \color{blue}{i \cdot y} \]

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. *-commutative N/A

       \[\leadsto \color{blue}{y \cdot i} + a \]

    3. lower-fma.f64 40.9

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, a\right)} \]

11. Simplified 40.9%

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, i, a\right)} \]
12. Add Preprocessing

Alternative 17: 24.3% accurate, 39.0× speedup

Math

\[\begin{array}{l} \\ y \cdot i \end{array} \]

FPCore

(FPCore (x y z t a b c i) :precision binary64 (* y i))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return 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 = 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 y * i;
}

Python

def code(x, y, z, t, a, b, c, i):
	return y * i

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(y * i)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = y * i;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(y * i), $MachinePrecision]

Derivation

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 y around inf

   \[\leadsto \color{blue}{i \cdot y} \]
4. Step-by-step derivation

   1. lower-*.f64 25.2

      \[\leadsto \color{blue}{i \cdot y} \]

5. Simplified 25.2%

   \[\leadsto \color{blue}{i \cdot y} \]

6. Final simplification 25.2%

   \[\leadsto y \cdot i \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024208 
(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)))