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

Percentage Accurate: 99.8% → 99.8%

Time: 15.8s

Alternatives: 20

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 2: 63.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(b - 0.5\right) \cdot \log c\\ t_2 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + t\_1\right) + y \cdot i\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+306}:\\ \;\;\;\;z + y \cdot i\\ \mathbf{elif}\;t\_2 \leq 10^{+39}:\\ \;\;\;\;a + \left(z + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(a + t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (- b 0.5) (log c)))
        (t_2 (+ (+ (+ (+ (+ (* x (log y)) z) t) a) t_1) (* y i))))
   (if (<= t_2 -2e+306)
     (+ z (* y i))
     (if (<= t_2 1e+39)
       (+ a (+ z (fma (log c) (+ b -0.5) t)))
       (+ (* y i) (+ a t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (b - 0.5) * log(c);
	double t_2 = (((((x * log(y)) + z) + t) + a) + t_1) + (y * i);
	double tmp;
	if (t_2 <= -2e+306) {
		tmp = z + (y * i);
	} else if (t_2 <= 1e+39) {
		tmp = a + (z + fma(log(c), (b + -0.5), t));
	} else {
		tmp = (y * i) + (a + t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(b - 0.5) * log(c))
	t_2 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + t_1) + Float64(y * i))
	tmp = 0.0
	if (t_2 <= -2e+306)
		tmp = Float64(z + Float64(y * i));
	elseif (t_2 <= 1e+39)
		tmp = Float64(a + Float64(z + fma(log(c), Float64(b + -0.5), t)));
	else
		tmp = Float64(Float64(y * i) + Float64(a + t_1));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+306], N[(z + N[(y * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+39], N[(a + N[(z + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * i), $MachinePrecision] + N[(a + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -2.00000000000000003e306

   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 z around inf

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

   5. Simplified 80.1%

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

   if -2.00000000000000003e306 < (+.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)) < 9.9999999999999994e38

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

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

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

      6. +-commutative N/A

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

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

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 84.0

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

   5. Simplified 84.0%

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

   6. Taylor expanded in i around 0

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

   8. Simplified 76.3%

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

   if 9.9999999999999994e38 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

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

   5. Simplified 51.7%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -2 \cdot 10^{+306}:\\ \;\;\;\;z + y \cdot i\\ \mathbf{elif}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq 10^{+39}:\\ \;\;\;\;a + \left(z + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(a + \left(b - 0.5\right) \cdot \log c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a + \left(t + \mathsf{fma}\left(i, y, z\right)\right)\\ t_2 := \left(b - 0.5\right) \cdot \log c\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+220}:\\ \;\;\;\;y \cdot i + b \cdot \log c\\ \mathbf{elif}\;t\_2 \leq -100:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 192:\\ \;\;\;\;a + \left(z + \mathsf{fma}\left(x, \log y, t\right)\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+162}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log c, b, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ a (+ t (fma i y z)))) (t_2 (* (- b 0.5) (log c))))
   (if (<= t_2 -5e+220)
     (+ (* y i) (* b (log c)))
     (if (<= t_2 -100.0)
       t_1
       (if (<= t_2 192.0)
         (+ a (+ z (fma x (log y) t)))
         (if (<= t_2 1e+162) t_1 (fma (log c) b a)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = a + (t + fma(i, y, z));
	double t_2 = (b - 0.5) * log(c);
	double tmp;
	if (t_2 <= -5e+220) {
		tmp = (y * i) + (b * log(c));
	} else if (t_2 <= -100.0) {
		tmp = t_1;
	} else if (t_2 <= 192.0) {
		tmp = a + (z + fma(x, log(y), t));
	} else if (t_2 <= 1e+162) {
		tmp = t_1;
	} else {
		tmp = fma(log(c), b, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(a + Float64(t + fma(i, y, z)))
	t_2 = Float64(Float64(b - 0.5) * log(c))
	tmp = 0.0
	if (t_2 <= -5e+220)
		tmp = Float64(Float64(y * i) + Float64(b * log(c)));
	elseif (t_2 <= -100.0)
		tmp = t_1;
	elseif (t_2 <= 192.0)
		tmp = Float64(a + Float64(z + fma(x, log(y), t)));
	elseif (t_2 <= 1e+162)
		tmp = t_1;
	else
		tmp = fma(log(c), b, a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(a + N[(t + N[(i * y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+220], N[(N[(y * i), $MachinePrecision] + N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -100.0], t$95$1, If[LessEqual[t$95$2, 192.0], N[(a + N[(z + N[(x * N[Log[y], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+162], t$95$1, N[(N[Log[c], $MachinePrecision] * b + a), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < -5.0000000000000002e220

   1. Initial program 99.5%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

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

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

      3. log-lowering-log.f64 82.2

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

   5. Simplified 82.2%

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

   if -5.0000000000000002e220 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < -100 or 192 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < 9.9999999999999994e161

   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 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

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

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

      6. +-commutative N/A

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

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

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 86.8

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

   5. Simplified 86.8%

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

   6. Taylor expanded in t around inf

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

   8. Simplified 77.5%

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

   if -100 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < 192

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

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

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

      2. associate-+r+ N/A

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

      3. cancel-sign-sub N/A

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

      4. log-rec N/A

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

      5. unsub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. associate-+l+ N/A

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

   5. Simplified 89.5%

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

   6. Taylor expanded in z around inf

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

   8. Simplified 82.9%

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

   if 9.9999999999999994e161 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))

   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 y around 0

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

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

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

      2. associate-+r+ N/A

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

      3. cancel-sign-sub N/A

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

      4. log-rec N/A

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

      5. unsub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. associate-+l+ N/A

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

   8. Simplified 59.7%

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

   9. Taylor expanded in b around inf

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

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

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

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

       5. *-commutative N/A

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

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

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

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

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

       8. mul-1-neg N/A

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

       9. neg-lowering-neg.f64 44.3

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

   11. Simplified 44.3%

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

   12. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

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

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

       4. log-lowering-log.f64 75.9

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

   14. Simplified 75.9%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(b - 0.5\right) \cdot \log c \leq -5 \cdot 10^{+220}:\\ \;\;\;\;y \cdot i + b \cdot \log c\\ \mathbf{elif}\;\left(b - 0.5\right) \cdot \log c \leq -100:\\ \;\;\;\;a + \left(t + \mathsf{fma}\left(i, y, z\right)\right)\\ \mathbf{elif}\;\left(b - 0.5\right) \cdot \log c \leq 192:\\ \;\;\;\;a + \left(z + \mathsf{fma}\left(x, \log y, t\right)\right)\\ \mathbf{elif}\;\left(b - 0.5\right) \cdot \log c \leq 10^{+162}:\\ \;\;\;\;a + \left(t + \mathsf{fma}\left(i, y, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log c, b, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\log c, b, a\right)\\ t_2 := a + \left(t + \mathsf{fma}\left(i, y, z\right)\right)\\ t_3 := \left(b - 0.5\right) \cdot \log c\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+197}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -100:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 192:\\ \;\;\;\;a + \left(z + \mathsf{fma}\left(x, \log y, t\right)\right)\\ \mathbf{elif}\;t\_3 \leq 10^{+162}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (log c) b a))
        (t_2 (+ a (+ t (fma i y z))))
        (t_3 (* (- b 0.5) (log c))))
   (if (<= t_3 -2e+197)
     t_1
     (if (<= t_3 -100.0)
       t_2
       (if (<= t_3 192.0)
         (+ a (+ z (fma x (log y) t)))
         (if (<= t_3 1e+162) t_2 t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(log(c), b, a);
	double t_2 = a + (t + fma(i, y, z));
	double t_3 = (b - 0.5) * log(c);
	double tmp;
	if (t_3 <= -2e+197) {
		tmp = t_1;
	} else if (t_3 <= -100.0) {
		tmp = t_2;
	} else if (t_3 <= 192.0) {
		tmp = a + (z + fma(x, log(y), t));
	} else if (t_3 <= 1e+162) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(log(c), b, a)
	t_2 = Float64(a + Float64(t + fma(i, y, z)))
	t_3 = Float64(Float64(b - 0.5) * log(c))
	tmp = 0.0
	if (t_3 <= -2e+197)
		tmp = t_1;
	elseif (t_3 <= -100.0)
		tmp = t_2;
	elseif (t_3 <= 192.0)
		tmp = Float64(a + Float64(z + fma(x, log(y), t)));
	elseif (t_3 <= 1e+162)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[c], $MachinePrecision] * b + a), $MachinePrecision]}, Block[{t$95$2 = N[(a + N[(t + N[(i * y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+197], t$95$1, If[LessEqual[t$95$3, -100.0], t$95$2, If[LessEqual[t$95$3, 192.0], N[(a + N[(z + N[(x * N[Log[y], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+162], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < -1.9999999999999999e197 or 9.9999999999999994e161 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))

   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 y around 0

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

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

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

      2. associate-+r+ N/A

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

      3. cancel-sign-sub N/A

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

      4. log-rec N/A

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

      5. unsub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. associate-+l+ N/A

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

   5. Simplified 90.3%

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

   6. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

   8. Simplified 55.7%

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

   9. Taylor expanded in b around inf

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

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

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

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

       5. *-commutative N/A

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

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

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

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

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

       8. mul-1-neg N/A

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

       9. neg-lowering-neg.f64 43.8

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

   11. Simplified 43.8%

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

   12. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

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

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

       4. log-lowering-log.f64 72.1

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

   14. Simplified 72.1%

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

   if -1.9999999999999999e197 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < -100 or 192 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < 9.9999999999999994e161

   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 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

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

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

      6. +-commutative N/A

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

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

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 86.5

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

   5. Simplified 86.5%

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

   6. Taylor expanded in t around inf

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

   8. Simplified 78.0%

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

   if -100 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < 192

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

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

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

      2. associate-+r+ N/A

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

      3. cancel-sign-sub N/A

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

      4. log-rec N/A

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

      5. unsub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. associate-+l+ N/A

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

   5. Simplified 89.5%

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

   6. Taylor expanded in z around inf

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

   8. Simplified 82.9%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(b - 0.5\right) \cdot \log c \leq -2 \cdot 10^{+197}:\\ \;\;\;\;\mathsf{fma}\left(\log c, b, a\right)\\ \mathbf{elif}\;\left(b - 0.5\right) \cdot \log c \leq -100:\\ \;\;\;\;a + \left(t + \mathsf{fma}\left(i, y, z\right)\right)\\ \mathbf{elif}\;\left(b - 0.5\right) \cdot \log c \leq 192:\\ \;\;\;\;a + \left(z + \mathsf{fma}\left(x, \log y, t\right)\right)\\ \mathbf{elif}\;\left(b - 0.5\right) \cdot \log c \leq 10^{+162}:\\ \;\;\;\;a + \left(t + \mathsf{fma}\left(i, y, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log c, b, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 40.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+25}:\\ \;\;\;\;z + a\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 (- INFINITY))
     (* y i)
     (if (<= t_1 -5e+25) (+ z a) (+ a (* y i))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y * i;
	} else if (t_1 <= -5e+25) {
		tmp = z + a;
	} else {
		tmp = a + (y * i);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = y * i;
	} else if (t_1 <= -5e+25) {
		tmp = z + a;
	} else {
		tmp = a + (y * i);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = y * i
	elif t_1 <= -5e+25:
		tmp = z + a
	else:
		tmp = a + (y * i)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y * i);
	elseif (t_1 <= -5e+25)
		tmp = Float64(z + a);
	else
		tmp = Float64(a + Float64(y * i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = y * i;
	elseif (t_1 <= -5e+25)
		tmp = z + a;
	else
		tmp = a + (y * i);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(y * i), $MachinePrecision], If[LessEqual[t$95$1, -5e+25], N[(z + a), $MachinePrecision], N[(a + N[(y * i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   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 y around inf

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

      1. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

   if -inf.0 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -5.00000000000000024e25

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

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

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

      6. +-commutative N/A

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

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

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 83.4

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

   5. Simplified 83.4%

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

   6. Taylor expanded in z around inf

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

   8. Simplified 34.3%

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

   if -5.00000000000000024e25 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

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

   5. Simplified 29.4%

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

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -\infty:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -5 \cdot 10^{+25}:\\ \;\;\;\;z + a\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+304}:\\ \;\;\;\;z + a\\ \mathbf{else}:\\ \;\;\;\;y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 (- INFINITY)) (* y i) (if (<= t_1 5e+304) (+ z a) (* y i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y * i;
	} else if (t_1 <= 5e+304) {
		tmp = z + a;
	} else {
		tmp = y * i;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = y * i;
	} else if (t_1 <= 5e+304) {
		tmp = z + a;
	} else {
		tmp = y * i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = y * i
	elif t_1 <= 5e+304:
		tmp = z + a
	else:
		tmp = y * i
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y * i);
	elseif (t_1 <= 5e+304)
		tmp = Float64(z + a);
	else
		tmp = Float64(y * i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = y * i;
	elseif (t_1 <= 5e+304)
		tmp = z + a;
	else
		tmp = y * i;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(y * i), $MachinePrecision], If[LessEqual[t$95$1, 5e+304], N[(z + a), $MachinePrecision], N[(y * i), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -inf.0 or 4.9999999999999997e304 < (+.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 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 y around inf

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

      1. *-lowering-*.f64 96.0

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

   5. Simplified 96.0%

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

   if -inf.0 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 4.9999999999999997e304

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

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

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

      6. +-commutative N/A

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

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

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 83.0

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

   5. Simplified 83.0%

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

   6. Taylor expanded in z around inf

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

   8. Simplified 32.3%

      \[\leadsto a + \color{blue}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -\infty:\\ \;\;\;\;y \cdot i\\ \mathbf{elif}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq 5 \cdot 10^{+304}:\\ \;\;\;\;z + a\\ \mathbf{else}:\\ \;\;\;\;y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(b - 0.5\right) \cdot \log c\\ \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + t\_1\right) + y \cdot i \leq -5 \cdot 10^{+25}:\\ \;\;\;\;y \cdot i + \left(z + t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(a + t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (- b 0.5) (log c))))
   (if (<= (+ (+ (+ (+ (+ (* x (log y)) z) t) a) t_1) (* y i)) -5e+25)
     (+ (* y i) (+ z t_1))
     (+ (* y i) (+ a t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (b - 0.5) * log(c);
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + t_1) + (y * i)) <= -5e+25) {
		tmp = (y * i) + (z + t_1);
	} else {
		tmp = (y * i) + (a + t_1);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (b - 0.5d0) * log(c)
    if (((((((x * log(y)) + z) + t) + a) + t_1) + (y * i)) <= (-5d+25)) then
        tmp = (y * i) + (z + t_1)
    else
        tmp = (y * i) + (a + t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (b - 0.5) * Math.log(c);
	double tmp;
	if (((((((x * Math.log(y)) + z) + t) + a) + t_1) + (y * i)) <= -5e+25) {
		tmp = (y * i) + (z + t_1);
	} else {
		tmp = (y * i) + (a + t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (b - 0.5) * math.log(c)
	tmp = 0
	if ((((((x * math.log(y)) + z) + t) + a) + t_1) + (y * i)) <= -5e+25:
		tmp = (y * i) + (z + t_1)
	else:
		tmp = (y * i) + (a + t_1)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(b - 0.5) * log(c))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + t_1) + Float64(y * i)) <= -5e+25)
		tmp = Float64(Float64(y * i) + Float64(z + t_1));
	else
		tmp = Float64(Float64(y * i) + Float64(a + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (b - 0.5) * log(c);
	tmp = 0.0;
	if (((((((x * log(y)) + z) + t) + a) + t_1) + (y * i)) <= -5e+25)
		tmp = (y * i) + (z + t_1);
	else
		tmp = (y * i) + (a + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], -5e+25], N[(N[(y * i), $MachinePrecision] + N[(z + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(y * i), $MachinePrecision] + N[(a + t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -5.00000000000000024e25

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 47.7%

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

   if -5.00000000000000024e25 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

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

   5. Simplified 53.6%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -5 \cdot 10^{+25}:\\ \;\;\;\;y \cdot i + \left(z + \left(b - 0.5\right) \cdot \log c\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot i + \left(a + \left(b - 0.5\right) \cdot \log c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\log c, b, a\right)\\ t_2 := \left(b - 0.5\right) \cdot \log c\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+197}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+162}:\\ \;\;\;\;a + \left(t + \mathsf{fma}\left(i, y, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (log c) b a)) (t_2 (* (- b 0.5) (log c))))
   (if (<= t_2 -2e+197) t_1 (if (<= t_2 1e+162) (+ a (+ t (fma i y 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(log(c), b, a);
	double t_2 = (b - 0.5) * log(c);
	double tmp;
	if (t_2 <= -2e+197) {
		tmp = t_1;
	} else if (t_2 <= 1e+162) {
		tmp = a + (t + fma(i, y, z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(log(c), b, a)
	t_2 = Float64(Float64(b - 0.5) * log(c))
	tmp = 0.0
	if (t_2 <= -2e+197)
		tmp = t_1;
	elseif (t_2 <= 1e+162)
		tmp = Float64(a + Float64(t + fma(i, y, z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[c], $MachinePrecision] * b + a), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+197], t$95$1, If[LessEqual[t$95$2, 1e+162], N[(a + N[(t + N[(i * y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < -1.9999999999999999e197 or 9.9999999999999994e161 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))

   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 y around 0

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

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

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

      2. associate-+r+ N/A

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

      3. cancel-sign-sub N/A

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

      4. log-rec N/A

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

      5. unsub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. associate-+l+ N/A

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

   5. Simplified 90.3%

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

   6. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

   8. Simplified 55.7%

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

   9. Taylor expanded in b around inf

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

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

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

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

       5. *-commutative N/A

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

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

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

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

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

       8. mul-1-neg N/A

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

       9. neg-lowering-neg.f64 43.8

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

   11. Simplified 43.8%

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

   12. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

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

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

       4. log-lowering-log.f64 72.1

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

   14. Simplified 72.1%

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

   if -1.9999999999999999e197 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < 9.9999999999999994e161

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

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

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

      6. +-commutative N/A

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

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

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 83.0

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

   5. Simplified 83.0%

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

   6. Taylor expanded in t around inf

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

   8. Simplified 75.0%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(b - 0.5\right) \cdot \log c \leq -2 \cdot 10^{+197}:\\ \;\;\;\;\mathsf{fma}\left(\log c, b, a\right)\\ \mathbf{elif}\;\left(b - 0.5\right) \cdot \log c \leq 10^{+162}:\\ \;\;\;\;a + \left(t + \mathsf{fma}\left(i, y, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log c, b, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \log c\\ t_2 := \left(b - 0.5\right) \cdot \log c\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+220}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+162}:\\ \;\;\;\;a + \left(t + \mathsf{fma}\left(i, y, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* b (log c))) (t_2 (* (- b 0.5) (log c))))
   (if (<= t_2 -5e+220) t_1 (if (<= t_2 1e+162) (+ a (+ t (fma i y 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 = b * log(c);
	double t_2 = (b - 0.5) * log(c);
	double tmp;
	if (t_2 <= -5e+220) {
		tmp = t_1;
	} else if (t_2 <= 1e+162) {
		tmp = a + (t + fma(i, y, z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(b * log(c))
	t_2 = Float64(Float64(b - 0.5) * log(c))
	tmp = 0.0
	if (t_2 <= -5e+220)
		tmp = t_1;
	elseif (t_2 <= 1e+162)
		tmp = Float64(a + Float64(t + fma(i, y, z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+220], t$95$1, If[LessEqual[t$95$2, 1e+162], N[(a + N[(t + N[(i * y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < -5.0000000000000002e220 or 9.9999999999999994e161 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))

   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 b around inf

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

      1. *-commutative N/A

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

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

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

      3. log-lowering-log.f64 68.5

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

   5. Simplified 68.5%

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

   if -5.0000000000000002e220 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < 9.9999999999999994e161

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

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

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

      6. +-commutative N/A

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

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

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 83.2

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

   5. Simplified 83.2%

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

   6. Taylor expanded in t around inf

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

   8. Simplified 74.7%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(b - 0.5\right) \cdot \log c \leq -5 \cdot 10^{+220}:\\ \;\;\;\;b \cdot \log c\\ \mathbf{elif}\;\left(b - 0.5\right) \cdot \log c \leq 10^{+162}:\\ \;\;\;\;a + \left(t + \mathsf{fma}\left(i, y, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \log c\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 37.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -50:\\ \;\;\;\;z + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;a + y \cdot i\\ \end{array} \end{array} \]

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))
      -50.0)
   (+ z (* y i))
   (+ a (* y i))))

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)) <= -50.0) {
		tmp = z + (y * i);
	} else {
		tmp = a + (y * i);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)) <= (-50.0d0)) then
        tmp = z + (y * i)
    else
        tmp = a + (y * i)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i)) <= -50.0) {
		tmp = z + (y * i);
	} else {
		tmp = a + (y * i);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)) <= -50.0:
		tmp = z + (y * i)
	else:
		tmp = a + (y * i)
	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)) <= -50.0)
		tmp = Float64(z + Float64(y * i));
	else
		tmp = Float64(a + Float64(y * i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -50.0)
		tmp = z + (y * i);
	else
		tmp = a + (y * i);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], -50.0], N[(z + N[(y * i), $MachinePrecision]), $MachinePrecision], N[(a + N[(y * i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -50

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 32.1%

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

   if -50 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

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

   5. Simplified 30.0%

      \[\leadsto \color{blue}{a} + y \cdot i \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 23.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -50:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;t + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
      -50.0)
   z
   (+ t a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -50.0) {
		tmp = z;
	} else {
		tmp = t + a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)) <= (-50.0d0)) then
        tmp = z
    else
        tmp = t + a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i)) <= -50.0) {
		tmp = z;
	} else {
		tmp = t + a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)) <= -50.0:
		tmp = z
	else:
		tmp = t + a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= -50.0)
		tmp = z;
	else
		tmp = Float64(t + a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -50.0)
		tmp = z;
	else
		tmp = t + a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], -50.0], z, N[(t + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -50

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 16.2%

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

   if -50 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

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

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

      6. +-commutative N/A

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

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

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 84.2

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

   5. Simplified 84.2%

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

   6. Taylor expanded in t around inf

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

   8. Simplified 26.4%

      \[\leadsto a + \color{blue}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 21.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -50:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;t + a\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 16.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -50:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
      -50.0)
   z
   a))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -50.0) {
		tmp = z;
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)) <= (-50.0d0)) then
        tmp = z
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i)) <= -50.0) {
		tmp = z;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)) <= -50.0:
		tmp = z
	else:
		tmp = a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= -50.0)
		tmp = z;
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -50.0)
		tmp = z;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], -50.0], z, a]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -50

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 16.2%

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

   if -50 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a} \]
   4. Step-by-step derivation

   5. Simplified 11.8%

      \[\leadsto \color{blue}{a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 92.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a + \left(\mathsf{fma}\left(\log c, b + -0.5, z\right) + \mathsf{fma}\left(x, \log y, t\right)\right)\\ \mathbf{if}\;x \leq -2.1 \cdot 10^{+111}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+87}:\\ \;\;\;\;a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ a (+ (fma (log c) (+ b -0.5) z) (fma x (log y) t)))))
   (if (<= x -2.1e+111)
     t_1
     (if (<= x 7e+87) (+ 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 = a + (fma(log(c), (b + -0.5), z) + fma(x, log(y), t));
	double tmp;
	if (x <= -2.1e+111) {
		tmp = t_1;
	} else if (x <= 7e+87) {
		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 = Float64(a + Float64(fma(log(c), Float64(b + -0.5), z) + fma(x, log(y), t)))
	tmp = 0.0
	if (x <= -2.1e+111)
		tmp = t_1;
	elseif (x <= 7e+87)
		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[(a + N[(N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + z), $MachinePrecision] + N[(x * N[Log[y], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.1e+111], t$95$1, If[LessEqual[x, 7e+87], 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 < -2.09999999999999995e111 or 6.99999999999999972e87 < x

   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 0

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

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

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

      2. associate-+r+ N/A

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

      3. cancel-sign-sub N/A

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

      4. log-rec N/A

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

      5. unsub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. associate-+l+ N/A

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

   5. Simplified 88.1%

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

   if -2.09999999999999995e111 < x < 6.99999999999999972e87

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

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

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

      6. +-commutative N/A

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

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

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 98.9

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

   5. Simplified 98.9%

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

Alternative 14: 90.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, \log y, t\right)\\ \mathbf{if}\;x \leq -3.3 \cdot 10^{+200}:\\ \;\;\;\;a + \left(t\_1 + b \cdot \log c\right)\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+160}:\\ \;\;\;\;a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a + \left(z + t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma x (log y) t)))
   (if (<= x -3.3e+200)
     (+ a (+ t_1 (* b (log c))))
     (if (<= x 1.3e+160)
       (+ a (+ (fma i y z) (fma (log c) (+ b -0.5) t)))
       (+ a (+ 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(x, log(y), t);
	double tmp;
	if (x <= -3.3e+200) {
		tmp = a + (t_1 + (b * log(c)));
	} else if (x <= 1.3e+160) {
		tmp = a + (fma(i, y, z) + fma(log(c), (b + -0.5), t));
	} else {
		tmp = a + (z + t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(x, log(y), t)
	tmp = 0.0
	if (x <= -3.3e+200)
		tmp = Float64(a + Float64(t_1 + Float64(b * log(c))));
	elseif (x <= 1.3e+160)
		tmp = Float64(a + Float64(fma(i, y, z) + fma(log(c), Float64(b + -0.5), t)));
	else
		tmp = Float64(a + Float64(z + t_1));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[x, -3.3e+200], N[(a + N[(t$95$1 + N[(b * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.3e+160], N[(a + N[(N[(i * y + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b + -0.5), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(z + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.3e200

   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 0

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

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

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

      2. associate-+r+ N/A

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

      3. cancel-sign-sub N/A

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

      4. log-rec N/A

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

      5. unsub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. associate-+l+ N/A

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

   5. Simplified 91.6%

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

   6. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

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

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

      3. log-lowering-log.f64 84.3

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

   8. Simplified 84.3%

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

   if -3.3e200 < x < 1.3e160

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

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

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

      6. +-commutative N/A

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

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

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 96.8

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

   5. Simplified 96.8%

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

   if 1.3e160 < x

   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 0

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

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

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

      2. associate-+r+ N/A

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

      3. cancel-sign-sub N/A

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

      4. log-rec N/A

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

      5. unsub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. associate-+l+ N/A

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

   5. Simplified 89.5%

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

   6. Taylor expanded in z around inf

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

   8. Simplified 86.0%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{+200}:\\ \;\;\;\;a + \left(\mathsf{fma}\left(x, \log y, t\right) + b \cdot \log c\right)\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+160}:\\ \;\;\;\;a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a + \left(z + \mathsf{fma}\left(x, \log y, t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 90.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a + \left(z + \mathsf{fma}\left(x, \log y, t\right)\right)\\ \mathbf{if}\;x \leq -4.7 \cdot 10^{+198}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+162}:\\ \;\;\;\;a + \left(\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ a (+ z (fma x (log y) t)))))
   (if (<= x -4.7e+198)
     t_1
     (if (<= x 6.8e+162)
       (+ 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 = a + (z + fma(x, log(y), t));
	double tmp;
	if (x <= -4.7e+198) {
		tmp = t_1;
	} else if (x <= 6.8e+162) {
		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 = Float64(a + Float64(z + fma(x, log(y), t)))
	tmp = 0.0
	if (x <= -4.7e+198)
		tmp = t_1;
	elseif (x <= 6.8e+162)
		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[(a + N[(z + N[(x * N[Log[y], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.7e+198], t$95$1, If[LessEqual[x, 6.8e+162], 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 < -4.7000000000000002e198 or 6.80000000000000006e162 < x

   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 0

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

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

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

      2. associate-+r+ N/A

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

      3. cancel-sign-sub N/A

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

      4. log-rec N/A

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

      5. unsub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. associate-+l+ N/A

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

   5. Simplified 90.4%

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

   6. Taylor expanded in z around inf

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

   8. Simplified 80.3%

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

   if -4.7000000000000002e198 < x < 6.80000000000000006e162

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

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

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

      6. +-commutative N/A

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

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

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 96.8

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

   5. Simplified 96.8%

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

Alternative 16: 72.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a + \left(z + \mathsf{fma}\left(x, \log y, t\right)\right)\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{+114}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+156}:\\ \;\;\;\;a + \left(z + \mathsf{fma}\left(\log c, b + -0.5, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ a (+ z (fma x (log y) t)))))
   (if (<= x -7.8e+114)
     t_1
     (if (<= x 1.65e+156) (+ a (+ 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 = a + (z + fma(x, log(y), t));
	double tmp;
	if (x <= -7.8e+114) {
		tmp = t_1;
	} else if (x <= 1.65e+156) {
		tmp = a + (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 = Float64(a + Float64(z + fma(x, log(y), t)))
	tmp = 0.0
	if (x <= -7.8e+114)
		tmp = t_1;
	elseif (x <= 1.65e+156)
		tmp = Float64(a + Float64(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[(a + N[(z + N[(x * N[Log[y], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.8e+114], t$95$1, If[LessEqual[x, 1.65e+156], N[(a + N[(z + 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 < -7.8000000000000001e114 or 1.6499999999999999e156 < x

   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 0

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

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

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

      2. associate-+r+ N/A

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

      3. cancel-sign-sub N/A

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

      4. log-rec N/A

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

      5. unsub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. associate-+l+ N/A

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

   5. Simplified 87.3%

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

   6. Taylor expanded in z around inf

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

   8. Simplified 77.3%

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

   if -7.8000000000000001e114 < x < 1.6499999999999999e156

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

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

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

      6. +-commutative N/A

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

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

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 98.3

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

   5. Simplified 98.3%

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

   6. Taylor expanded in i around 0

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

   8. Simplified 79.6%

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

Alternative 17: 71.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -4.3 \cdot 10^{+198}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+243}:\\ \;\;\;\;a + \left(t + \mathsf{fma}\left(i, y, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -4.3e+198) t_1 (if (<= x 5.8e+243) (+ a (+ t (fma i y 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 = x * log(y);
	double tmp;
	if (x <= -4.3e+198) {
		tmp = t_1;
	} else if (x <= 5.8e+243) {
		tmp = a + (t + fma(i, y, z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -4.3e+198)
		tmp = t_1;
	elseif (x <= 5.8e+243)
		tmp = Float64(a + Float64(t + fma(i, y, z)));
	else
		tmp = t_1;
	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]}, If[LessEqual[x, -4.3e+198], t$95$1, If[LessEqual[x, 5.8e+243], N[(a + N[(t + N[(i * y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.29999999999999982e198 or 5.80000000000000013e243 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

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

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

      2. log-lowering-log.f64 67.2

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

   5. Simplified 67.2%

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

   if -4.29999999999999982e198 < x < 5.80000000000000013e243

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

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

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

      6. +-commutative N/A

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

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

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lowering-+.f64 94.7

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

   5. Simplified 94.7%

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

   6. Taylor expanded in t around inf

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

   8. Simplified 72.7%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{+198}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+243}:\\ \;\;\;\;a + \left(t + \mathsf{fma}\left(i, y, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 67.3% accurate, 18.0× speedup

Math

\[\begin{array}{l} \\ a + \left(t + \mathsf{fma}\left(i, y, z\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

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

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

   2. +-commutative N/A

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

   3. associate-+r+ N/A

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

   4. associate-+l+ N/A

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

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

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

   6. +-commutative N/A

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

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

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

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

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

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

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. +-lowering-+.f64 84.7

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

5. Simplified 84.7%

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

6. Taylor expanded in t around inf

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

8. Simplified 64.5%

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

9. Final simplification 64.5%

   \[\leadsto a + \left(t + \mathsf{fma}\left(i, y, z\right)\right) \]
10. Add Preprocessing

Alternative 19: 30.9% accurate, 58.5× speedup

Math

\[\begin{array}{l} \\ z + a \end{array} \]

FPCore

(FPCore (x y z t a b c i) :precision binary64 (+ z a))

C

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

Fortran

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

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return z + a;
}

Python

def code(x, y, z, t, a, b, c, i):
	return z + a

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(z + a), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

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

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

   2. +-commutative N/A

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

   3. associate-+r+ N/A

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

   4. associate-+l+ N/A

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

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

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

   6. +-commutative N/A

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

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

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

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

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

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

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. +-lowering-+.f64 84.7

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

5. Simplified 84.7%

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

6. Taylor expanded in z around inf

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

8. Simplified 29.4%

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

9. Final simplification 29.4%

   \[\leadsto z + a \]
10. Add Preprocessing

Alternative 20: 16.1% accurate, 234.0× speedup

Math

\[\begin{array}{l} \\ a \end{array} \]

FPCore

(FPCore (x y z t a b c i) :precision binary64 a)

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return a;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = a
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return a;
}

Python

def code(x, y, z, t, a, b, c, i):
	return a

Julia

function code(x, y, z, t, a, b, c, i)
	return a
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = a;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := a

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a} \]
4. Step-by-step derivation

5. Simplified 14.4%

   \[\leadsto \color{blue}{a} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(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)))