Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A

Percentage Accurate: 99.9% → 99.9%

Time: 13.2s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (x y z t) :precision binary64 (+ (- (- (* x (log y)) y) z) (log t)))

C

double code(double x, double y, double z, double t) {
	return (((x * log(y)) - y) - z) + log(t);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x * log(y)) - y) - z) + log(t)
end function

Java

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

Python

def code(x, y, z, t):
	return (((x * math.log(y)) - y) - z) + math.log(t)

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * log(y)) - y) - z) + log(t))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * log(y)) - y) - z) + log(t);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision] + N[Log[t], $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t) :precision binary64 (+ (- (- (* x (log y)) y) z) (log t)))

C

double code(double x, double y, double z, double t) {
	return (((x * log(y)) - y) - z) + log(t);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x * log(y)) - y) - z) + log(t)
end function

Java

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

Python

def code(x, y, z, t):
	return (((x * math.log(y)) - y) - z) + math.log(t)

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * log(y)) - y) - z) + log(t))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * log(y)) - y) - z) + log(t);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision] + N[Log[t], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t) :precision binary64 (+ (- (- (* x (log y)) y) z) (log t)))

C

double code(double x, double y, double z, double t) {
	return (((x * log(y)) - y) - z) + log(t);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x * log(y)) - y) - z) + log(t)
end function

Java

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

Python

def code(x, y, z, t):
	return (((x * math.log(y)) - y) - z) + math.log(t)

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * log(y)) - y) - z) + log(t))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * log(y)) - y) - z) + log(t);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision] + N[Log[t], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

Alternative 2: 68.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \log y - y\right) - z\\ \mathbf{if}\;t\_1 \leq -500:\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{elif}\;t\_1 \leq 40000000:\\ \;\;\;\;\log t\\ \mathbf{else}:\\ \;\;\;\;y - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (- (* x (log y)) y) z)))
   (if (<= t_1 -500.0) (- (- y) z) (if (<= t_1 40000000.0) (log t) (- y z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((x * log(y)) - y) - z;
	double tmp;
	if (t_1 <= -500.0) {
		tmp = -y - z;
	} else if (t_1 <= 40000000.0) {
		tmp = log(t);
	} else {
		tmp = y - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((x * log(y)) - y) - z
    if (t_1 <= (-500.0d0)) then
        tmp = -y - z
    else if (t_1 <= 40000000.0d0) then
        tmp = log(t)
    else
        tmp = y - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((x * Math.log(y)) - y) - z;
	double tmp;
	if (t_1 <= -500.0) {
		tmp = -y - z;
	} else if (t_1 <= 40000000.0) {
		tmp = Math.log(t);
	} else {
		tmp = y - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((x * math.log(y)) - y) - z
	tmp = 0
	if t_1 <= -500.0:
		tmp = -y - z
	elif t_1 <= 40000000.0:
		tmp = math.log(t)
	else:
		tmp = y - z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(x * log(y)) - y) - z)
	tmp = 0.0
	if (t_1 <= -500.0)
		tmp = Float64(Float64(-y) - z);
	elseif (t_1 <= 40000000.0)
		tmp = log(t);
	else
		tmp = Float64(y - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((x * log(y)) - y) - z;
	tmp = 0.0;
	if (t_1 <= -500.0)
		tmp = -y - z;
	elseif (t_1 <= 40000000.0)
		tmp = log(t);
	else
		tmp = y - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$1, -500.0], N[((-y) - z), $MachinePrecision], If[LessEqual[t$95$1, 40000000.0], N[Log[t], $MachinePrecision], N[(y - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) < -500

   1. Initial program 99.8%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}}}} \]

      5. flip3-+ N/A

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf

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

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

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

      2. log-lowering-log.f64 98.1

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

   7. Simplified 98.1%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \left(y + z\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \color{blue}{-1 \cdot y + -1 \cdot z} \]

      2. mul-1-neg N/A

         \[\leadsto -1 \cdot y + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{-1 \cdot y - z} \]

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

         \[\leadsto \color{blue}{-1 \cdot y - z} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} - z \]

      6. neg-lowering-neg.f64 71.8

         \[\leadsto \color{blue}{\left(-y\right)} - z \]

   10. Simplified 71.8%

       \[\leadsto \color{blue}{\left(-y\right) - z} \]

   if -500 < (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) < 4e7

   1. Initial program 99.9%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-1 \cdot z} + \log t \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + \log t \]

      2. neg-lowering-neg.f64 92.0

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

   5. Simplified 92.0%

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

   6. Taylor expanded in z around 0

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

      1. log-lowering-log.f64 89.6

         \[\leadsto \color{blue}{\log t} \]

   8. Simplified 89.6%

      \[\leadsto \color{blue}{\log t} \]

   if 4e7 < (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z)

   1. Initial program 99.7%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}}}} \]

      5. flip3-+ N/A

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in x around inf

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

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

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

      2. log-lowering-log.f64 99.2

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

   7. Simplified 99.2%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \left(y + z\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \color{blue}{-1 \cdot y + -1 \cdot z} \]

      2. mul-1-neg N/A

         \[\leadsto -1 \cdot y + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{-1 \cdot y - z} \]

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

         \[\leadsto \color{blue}{-1 \cdot y - z} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} - z \]

      6. neg-lowering-neg.f64 50.9

         \[\leadsto \color{blue}{\left(-y\right)} - z \]

   10. Simplified 50.9%

       \[\leadsto \color{blue}{\left(-y\right) - z} \]
   11. Step-by-step derivation

       1. neg-sub0 N/A

          \[\leadsto \color{blue}{\left(0 - y\right)} - z \]

       2. flip3-- N/A

          \[\leadsto \color{blue}{\frac{{0}^{3} - {y}^{3}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}} - z \]

       3. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{0} - {y}^{3}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} - z \]

       4. cube-unmult N/A

          \[\leadsto \frac{0 - \color{blue}{y \cdot \left(y \cdot y\right)}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} - z \]

       5. neg-sub0 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(y \cdot \left(y \cdot y\right)\right)}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} - z \]

       6. cube-unmult N/A

          \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{y}^{3}}\right)}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} - z \]

       7. cube-neg N/A

          \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{3}}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} - z \]

       8. sqr-pow N/A

          \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(y\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} - z \]

       9. unpow-prod-down N/A

          \[\leadsto \frac{\color{blue}{{\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} - z \]

       10. sqr-neg N/A

           \[\leadsto \frac{{\color{blue}{\left(y \cdot y\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} - z \]

       11. unpow-prod-down N/A

           \[\leadsto \frac{\color{blue}{{y}^{\left(\frac{3}{2}\right)} \cdot {y}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} - z \]

       12. sqr-pow N/A

           \[\leadsto \frac{\color{blue}{{y}^{3}}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} - z \]

       13. metadata-eval N/A

           \[\leadsto \frac{{y}^{3}}{\color{blue}{0} + \left(y \cdot y + 0 \cdot y\right)} - z \]

       14. mul0-lft N/A

           \[\leadsto \frac{{y}^{3}}{0 + \left(y \cdot y + \color{blue}{0}\right)} - z \]

       15. metadata-eval N/A

           \[\leadsto \frac{{y}^{3}}{0 + \left(y \cdot y + \color{blue}{\left(\mathsf{neg}\left(0\right)\right)}\right)} - z \]

       16. mul0-lft N/A

           \[\leadsto \frac{{y}^{3}}{0 + \left(y \cdot y + \left(\mathsf{neg}\left(\color{blue}{0 \cdot \left(\mathsf{neg}\left(y\right)\right)}\right)\right)\right)} - z \]

       17. sub-neg N/A

           \[\leadsto \frac{{y}^{3}}{0 + \color{blue}{\left(y \cdot y - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}} - z \]

       18. +-lft-identity N/A

           \[\leadsto \frac{{y}^{3}}{\color{blue}{y \cdot y - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)}} - z \]

       19. mul0-lft N/A

           \[\leadsto \frac{{y}^{3}}{y \cdot y - \color{blue}{0}} - z \]

       20. --rgt-identity N/A

           \[\leadsto \frac{{y}^{3}}{\color{blue}{y \cdot y}} - z \]

       21. pow2 N/A

           \[\leadsto \frac{{y}^{3}}{\color{blue}{{y}^{2}}} - z \]

       22. pow-div N/A

           \[\leadsto \color{blue}{{y}^{\left(3 - 2\right)}} - z \]

       23. metadata-eval N/A

           \[\leadsto {y}^{\color{blue}{1}} - z \]

       24. unpow1 N/A

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

   12. Applied egg-rr 51.8%

       \[\leadsto \color{blue}{y - z} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 79.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t\_1 - y\\ \mathbf{if}\;t\_2 \leq -500:\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+28}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- t_1 y)))
   (if (<= t_2 -500.0) (- (- y) z) (if (<= t_2 2e+28) (- (log t) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = t_1 - y;
	double tmp;
	if (t_2 <= -500.0) {
		tmp = -y - z;
	} else if (t_2 <= 2e+28) {
		tmp = log(t) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * log(y)
    t_2 = t_1 - y
    if (t_2 <= (-500.0d0)) then
        tmp = -y - z
    else if (t_2 <= 2d+28) then
        tmp = log(t) - z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double t_2 = t_1 - y;
	double tmp;
	if (t_2 <= -500.0) {
		tmp = -y - z;
	} else if (t_2 <= 2e+28) {
		tmp = Math.log(t) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = t_1 - y
	tmp = 0
	if t_2 <= -500.0:
		tmp = -y - z
	elif t_2 <= 2e+28:
		tmp = math.log(t) - z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(t_1 - y)
	tmp = 0.0
	if (t_2 <= -500.0)
		tmp = Float64(Float64(-y) - z);
	elseif (t_2 <= 2e+28)
		tmp = Float64(log(t) - z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	t_2 = t_1 - y;
	tmp = 0.0;
	if (t_2 <= -500.0)
		tmp = -y - z;
	elseif (t_2 <= 2e+28)
		tmp = log(t) - z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - y), $MachinePrecision]}, If[LessEqual[t$95$2, -500.0], N[((-y) - z), $MachinePrecision], If[LessEqual[t$95$2, 2e+28], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -500

   1. Initial program 99.8%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}}}} \]

      5. flip3-+ N/A

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf

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

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

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

      2. log-lowering-log.f64 98.0

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

   7. Simplified 98.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \left(y + z\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \color{blue}{-1 \cdot y + -1 \cdot z} \]

      2. mul-1-neg N/A

         \[\leadsto -1 \cdot y + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{-1 \cdot y - z} \]

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

         \[\leadsto \color{blue}{-1 \cdot y - z} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} - z \]

      6. neg-lowering-neg.f64 69.8

         \[\leadsto \color{blue}{\left(-y\right)} - z \]

   10. Simplified 69.8%

       \[\leadsto \color{blue}{\left(-y\right) - z} \]

   if -500 < (-.f64 (*.f64 x (log.f64 y)) y) < 1.99999999999999992e28

   1. Initial program 99.9%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-1 \cdot z} + \log t \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + \log t \]

      2. neg-lowering-neg.f64 94.6

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

   5. Simplified 94.6%

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\log t + \left(\mathsf{neg}\left(z\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\log t - z} \]

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

         \[\leadsto \color{blue}{\log t - z} \]

      4. log-lowering-log.f64 94.6

         \[\leadsto \color{blue}{\log t} - z \]

   7. Applied egg-rr 94.6%

      \[\leadsto \color{blue}{\log t - z} \]

   if 1.99999999999999992e28 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.5%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   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 86.2

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

   5. Simplified 86.2%

      \[\leadsto \color{blue}{x \cdot \log y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\frac{1}{x \cdot \log y - \left(y + z\right)}}\\ \mathbf{if}\;z \leq -6500000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 120:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, \log t - y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 1.0 (/ 1.0 (- (* x (log y)) (+ y z))))))
   (if (<= z -6500000.0)
     t_1
     (if (<= z 120.0) (fma x (log y) (- (log t) y)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / (1.0 / ((x * log(y)) - (y + z)));
	double tmp;
	if (z <= -6500000.0) {
		tmp = t_1;
	} else if (z <= 120.0) {
		tmp = fma(x, log(y), (log(t) - y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(1.0 / Float64(1.0 / Float64(Float64(x * log(y)) - Float64(y + z))))
	tmp = 0.0
	if (z <= -6500000.0)
		tmp = t_1;
	elseif (z <= 120.0)
		tmp = fma(x, log(y), Float64(log(t) - y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 / N[(1.0 / N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6500000.0], t$95$1, If[LessEqual[z, 120.0], N[(x * N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.5e6 or 120 < z

   1. Initial program 99.9%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}}}} \]

      5. flip3-+ N/A

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf

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

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

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

      2. log-lowering-log.f64 99.6

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

   7. Simplified 99.6%

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

   if -6.5e6 < z < 120

   1. Initial program 99.7%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

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

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

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

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

      6. log-lowering-log.f64 99.1

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

   5. Simplified 99.1%

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

Alternative 5: 69.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot \log y - y \leq -500:\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\log t - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- (* x (log y)) y) -500.0) (- (- y) z) (- (log t) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * log(y)) - y) <= -500.0) {
		tmp = -y - z;
	} else {
		tmp = log(t) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x * log(y)) - y) <= (-500.0d0)) then
        tmp = -y - z
    else
        tmp = log(t) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * Math.log(y)) - y) <= -500.0) {
		tmp = -y - z;
	} else {
		tmp = Math.log(t) - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x * math.log(y)) - y) <= -500.0:
		tmp = -y - z
	else:
		tmp = math.log(t) - z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x * log(y)) - y) <= -500.0)
		tmp = Float64(Float64(-y) - z);
	else
		tmp = Float64(log(t) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x * log(y)) - y) <= -500.0)
		tmp = -y - z;
	else
		tmp = log(t) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision], -500.0], N[((-y) - z), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -500

   1. Initial program 99.8%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}}}} \]

      5. flip3-+ N/A

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf

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

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

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

      2. log-lowering-log.f64 98.0

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

   7. Simplified 98.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \left(y + z\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \color{blue}{-1 \cdot y + -1 \cdot z} \]

      2. mul-1-neg N/A

         \[\leadsto -1 \cdot y + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{-1 \cdot y - z} \]

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

         \[\leadsto \color{blue}{-1 \cdot y - z} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} - z \]

      6. neg-lowering-neg.f64 69.8

         \[\leadsto \color{blue}{\left(-y\right)} - z \]

   10. Simplified 69.8%

       \[\leadsto \color{blue}{\left(-y\right) - z} \]

   if -500 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.8%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-1 \cdot z} + \log t \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + \log t \]

      2. neg-lowering-neg.f64 68.4

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

   5. Simplified 68.4%

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\log t + \left(\mathsf{neg}\left(z\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\log t - z} \]

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

         \[\leadsto \color{blue}{\log t - z} \]

      4. log-lowering-log.f64 68.4

         \[\leadsto \color{blue}{\log t} - z \]

   7. Applied egg-rr 68.4%

      \[\leadsto \color{blue}{\log t - z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.2:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, \log t - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, \frac{\log y}{y}, -1 - \frac{z}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 2.2)
   (fma x (log y) (- (log t) z))
   (* y (fma x (/ (log y) y) (- -1.0 (/ z y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.2) {
		tmp = fma(x, log(y), (log(t) - z));
	} else {
		tmp = y * fma(x, (log(y) / y), (-1.0 - (z / y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.2)
		tmp = fma(x, log(y), Float64(log(t) - z));
	else
		tmp = Float64(y * fma(x, Float64(log(y) / y), Float64(-1.0 - Float64(z / y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 2.2], N[(x * N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x * N[(N[Log[y], $MachinePrecision] / y), $MachinePrecision] + N[(-1.0 - N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.2000000000000002

   1. Initial program 99.8%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

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

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

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

      6. unsub-neg N/A

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

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

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

      8. log-lowering-log.f64 99.3

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

   5. Simplified 99.3%

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

   if 2.2000000000000002 < y

   1. Initial program 99.9%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}}}} \]

      5. flip3-+ N/A

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf

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

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

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

      2. log-lowering-log.f64 98.3

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

   7. Simplified 98.3%

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

   8. Taylor expanded in y around inf

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

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

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

      2. associate--r+ N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. associate--l+ N/A

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

      6. associate-*r/ N/A

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

      7. mul-1-neg N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. associate-/l* N/A

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

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

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

      15. /-lowering-/.f64 N/A

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

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

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

      17. --lowering--.f64 N/A

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

   10. Simplified 98.4%

       \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(x, \frac{\log y}{y}, -1 - \frac{z}{y}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 99.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(\log y - \frac{y + z}{x}\right)\\ \mathbf{if}\;x \leq -1.2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.0085:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (- (log y) (/ (+ y z) x)))))
   (if (<= x -1.2) t_1 (if (<= x 0.0085) (- (log t) (+ y z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (log(y) - ((y + z) / x));
	double tmp;
	if (x <= -1.2) {
		tmp = t_1;
	} else if (x <= 0.0085) {
		tmp = log(t) - (y + z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (log(y) - ((y + z) / x))
    if (x <= (-1.2d0)) then
        tmp = t_1
    else if (x <= 0.0085d0) then
        tmp = log(t) - (y + z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * (Math.log(y) - ((y + z) / x));
	double tmp;
	if (x <= -1.2) {
		tmp = t_1;
	} else if (x <= 0.0085) {
		tmp = Math.log(t) - (y + z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (math.log(y) - ((y + z) / x))
	tmp = 0
	if x <= -1.2:
		tmp = t_1
	elif x <= 0.0085:
		tmp = math.log(t) - (y + z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(log(y) - Float64(Float64(y + z) / x)))
	tmp = 0.0
	if (x <= -1.2)
		tmp = t_1;
	elseif (x <= 0.0085)
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (log(y) - ((y + z) / x));
	tmp = 0.0;
	if (x <= -1.2)
		tmp = t_1;
	elseif (x <= 0.0085)
		tmp = log(t) - (y + z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(N[Log[y], $MachinePrecision] - N[(N[(y + z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.2], t$95$1, If[LessEqual[x, 0.0085], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.19999999999999996 or 0.0085000000000000006 < x

   1. Initial program 99.7%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}}}} \]

      5. flip3-+ N/A

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in x around inf

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

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

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

      2. log-lowering-log.f64 98.4

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

   7. Simplified 98.4%

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

   8. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. distribute-lft-in N/A

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

      3. mul-1-neg N/A

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

      4. sub-neg N/A

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

      5. div-sub N/A

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

      6. associate-*r/ N/A

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

      7. associate--l+ N/A

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

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

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

      9. associate--l+ N/A

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

      10. associate-*r/ N/A

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

      11. div-sub N/A

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

      12. sub-neg N/A

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

      13. mul-1-neg N/A

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

      14. distribute-lft-in N/A

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

      15. associate-*r/ N/A

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

      16. mul-1-neg N/A

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

      17. unsub-neg N/A

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

      18. --lowering--.f64 N/A

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

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

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

   10. Simplified 98.5%

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

   if -1.19999999999999996 < x < 0.0085000000000000006

   1. Initial program 100.0%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. +-lowering-+.f64 99.1

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

   5. Simplified 99.1%

      \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 90.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, -y\right)\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+36}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.7e+27)
   (fma x (log y) (- y))
   (if (<= x 6.2e+36) (- (log t) (+ y z)) (fma x (log y) (- z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.7e+27) {
		tmp = fma(x, log(y), -y);
	} else if (x <= 6.2e+36) {
		tmp = log(t) - (y + z);
	} else {
		tmp = fma(x, log(y), -z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.7e+27)
		tmp = fma(x, log(y), Float64(-y));
	elseif (x <= 6.2e+36)
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = fma(x, log(y), Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.7e+27], N[(x * N[Log[y], $MachinePrecision] + (-y)), $MachinePrecision], If[LessEqual[x, 6.2e+36], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], N[(x * N[Log[y], $MachinePrecision] + (-z)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.7e27

   1. Initial program 99.7%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}}}} \]

      5. flip3-+ N/A

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around inf

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

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

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

      2. log-lowering-log.f64 99.6

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

   7. Simplified 99.6%

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

   8. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

         \[\leadsto x \cdot \log y + \color{blue}{-1 \cdot y} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, -1 \cdot y\right)} \]

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

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\log y}, -1 \cdot y\right) \]

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 89.8

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

   10. Simplified 89.8%

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

   if -1.7e27 < x < 6.1999999999999999e36

   1. Initial program 99.9%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. +-lowering-+.f64 96.3

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

   5. Simplified 96.3%

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

   if 6.1999999999999999e36 < x

   1. Initial program 99.6%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

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

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

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

      6. unsub-neg N/A

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

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

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

      8. log-lowering-log.f64 88.1

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

   5. Simplified 88.1%

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

   6. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 88.1

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

   8. Simplified 88.1%

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

Alternative 9: 90.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, \log y, -y\right)\\ \mathbf{if}\;x \leq -2 \cdot 10^{+32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+38}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma x (log y) (- y))))
   (if (<= x -2e+32) t_1 (if (<= x 2.3e+38) (- (log t) (+ y z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(x, log(y), -y);
	double tmp;
	if (x <= -2e+32) {
		tmp = t_1;
	} else if (x <= 2.3e+38) {
		tmp = log(t) - (y + z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(x, log(y), Float64(-y))
	tmp = 0.0
	if (x <= -2e+32)
		tmp = t_1;
	elseif (x <= 2.3e+38)
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision] + (-y)), $MachinePrecision]}, If[LessEqual[x, -2e+32], t$95$1, If[LessEqual[x, 2.3e+38], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.00000000000000011e32 or 2.3000000000000001e38 < x

   1. Initial program 99.7%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}}}} \]

      5. flip3-+ N/A

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in x around inf

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

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

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

      2. log-lowering-log.f64 99.5

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

   7. Simplified 99.5%

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

   8. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

         \[\leadsto x \cdot \log y + \color{blue}{-1 \cdot y} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, -1 \cdot y\right)} \]

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

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\log y}, -1 \cdot y\right) \]

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 85.2

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

   10. Simplified 85.2%

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

   if -2.00000000000000011e32 < x < 2.3000000000000001e38

   1. Initial program 99.9%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. +-lowering-+.f64 96.4

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

   5. Simplified 96.4%

      \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 83.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -1.95 \cdot 10^{+123}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+38}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -1.95e+123) t_1 (if (<= x 8.5e+38) (- (log t) (+ y z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -1.95e+123) {
		tmp = t_1;
	} else if (x <= 8.5e+38) {
		tmp = log(t) - (y + z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if (x <= (-1.95d+123)) then
        tmp = t_1
    else if (x <= 8.5d+38) then
        tmp = log(t) - (y + z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double tmp;
	if (x <= -1.95e+123) {
		tmp = t_1;
	} else if (x <= 8.5e+38) {
		tmp = Math.log(t) - (y + z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -1.95e+123:
		tmp = t_1
	elif x <= 8.5e+38:
		tmp = math.log(t) - (y + z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -1.95e+123)
		tmp = t_1;
	elseif (x <= 8.5e+38)
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (x <= -1.95e+123)
		tmp = t_1;
	elseif (x <= 8.5e+38)
		tmp = log(t) - (y + z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.95e+123], t$95$1, If[LessEqual[x, 8.5e+38], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.94999999999999996e123 or 8.4999999999999997e38 < x

   1. Initial program 99.6%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   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 75.1

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

   5. Simplified 75.1%

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

   if -1.94999999999999996e123 < x < 8.4999999999999997e38

   1. Initial program 99.9%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. +-lowering-+.f64 94.0

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

   5. Simplified 94.0%

      \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 48.5% accurate, 21.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.1 \cdot 10^{+51}:\\ \;\;\;\;y - z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 5.1e+51) (- y z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 5.1e+51) {
		tmp = y - z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 5.1d+51) then
        tmp = y - z
    else
        tmp = -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 5.1e+51) {
		tmp = y - z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 5.1e+51:
		tmp = y - z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 5.1e+51)
		tmp = Float64(y - z);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 5.1e+51)
		tmp = y - z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 5.1e+51], N[(y - z), $MachinePrecision], (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 5.1000000000000001e51

   1. Initial program 99.8%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}}}} \]

      5. flip3-+ N/A

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in x around inf

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

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

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

      2. log-lowering-log.f64 79.2

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

   7. Simplified 79.2%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \left(y + z\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \color{blue}{-1 \cdot y + -1 \cdot z} \]

      2. mul-1-neg N/A

         \[\leadsto -1 \cdot y + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{-1 \cdot y - z} \]

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

         \[\leadsto \color{blue}{-1 \cdot y - z} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} - z \]

      6. neg-lowering-neg.f64 41.3

         \[\leadsto \color{blue}{\left(-y\right)} - z \]

   10. Simplified 41.3%

       \[\leadsto \color{blue}{\left(-y\right) - z} \]
   11. Step-by-step derivation

       1. neg-sub0 N/A

          \[\leadsto \color{blue}{\left(0 - y\right)} - z \]

       2. flip3-- N/A

          \[\leadsto \color{blue}{\frac{{0}^{3} - {y}^{3}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}} - z \]

       3. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{0} - {y}^{3}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} - z \]

       4. cube-unmult N/A

          \[\leadsto \frac{0 - \color{blue}{y \cdot \left(y \cdot y\right)}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} - z \]

       5. neg-sub0 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(y \cdot \left(y \cdot y\right)\right)}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} - z \]

       6. cube-unmult N/A

          \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{y}^{3}}\right)}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} - z \]

       7. cube-neg N/A

          \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{3}}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} - z \]

       8. sqr-pow N/A

          \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(y\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} - z \]

       9. unpow-prod-down N/A

          \[\leadsto \frac{\color{blue}{{\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} - z \]

       10. sqr-neg N/A

           \[\leadsto \frac{{\color{blue}{\left(y \cdot y\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} - z \]

       11. unpow-prod-down N/A

           \[\leadsto \frac{\color{blue}{{y}^{\left(\frac{3}{2}\right)} \cdot {y}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} - z \]

       12. sqr-pow N/A

           \[\leadsto \frac{\color{blue}{{y}^{3}}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} - z \]

       13. metadata-eval N/A

           \[\leadsto \frac{{y}^{3}}{\color{blue}{0} + \left(y \cdot y + 0 \cdot y\right)} - z \]

       14. mul0-lft N/A

           \[\leadsto \frac{{y}^{3}}{0 + \left(y \cdot y + \color{blue}{0}\right)} - z \]

       15. metadata-eval N/A

           \[\leadsto \frac{{y}^{3}}{0 + \left(y \cdot y + \color{blue}{\left(\mathsf{neg}\left(0\right)\right)}\right)} - z \]

       16. mul0-lft N/A

           \[\leadsto \frac{{y}^{3}}{0 + \left(y \cdot y + \left(\mathsf{neg}\left(\color{blue}{0 \cdot \left(\mathsf{neg}\left(y\right)\right)}\right)\right)\right)} - z \]

       17. sub-neg N/A

           \[\leadsto \frac{{y}^{3}}{0 + \color{blue}{\left(y \cdot y - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}} - z \]

       18. +-lft-identity N/A

           \[\leadsto \frac{{y}^{3}}{\color{blue}{y \cdot y - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)}} - z \]

       19. mul0-lft N/A

           \[\leadsto \frac{{y}^{3}}{y \cdot y - \color{blue}{0}} - z \]

       20. --rgt-identity N/A

           \[\leadsto \frac{{y}^{3}}{\color{blue}{y \cdot y}} - z \]

       21. pow2 N/A

           \[\leadsto \frac{{y}^{3}}{\color{blue}{{y}^{2}}} - z \]

       22. pow-div N/A

           \[\leadsto \color{blue}{{y}^{\left(3 - 2\right)}} - z \]

       23. metadata-eval N/A

           \[\leadsto {y}^{\color{blue}{1}} - z \]

       24. unpow1 N/A

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

   12. Applied egg-rr 39.3%

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

   if 5.1000000000000001e51 < y

   1. Initial program 99.9%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-1 \cdot y} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(y\right)} \]

      2. neg-lowering-neg.f64 74.0

         \[\leadsto \color{blue}{-y} \]

   5. Simplified 74.0%

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

Alternative 12: 48.6% accurate, 23.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{+49}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 3.6e+49) (- z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.6e+49) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 3.6d+49) then
        tmp = -z
    else
        tmp = -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.6e+49) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 3.6e+49:
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3.6e+49)
		tmp = Float64(-z);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 3.6e+49)
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 3.6e+49], (-z), (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 3.59999999999999996e49

   1. Initial program 99.8%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-1 \cdot z} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. neg-lowering-neg.f64 39.2

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

   5. Simplified 39.2%

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

   if 3.59999999999999996e49 < y

   1. Initial program 99.9%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-1 \cdot y} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(y\right)} \]

      2. neg-lowering-neg.f64 74.0

         \[\leadsto \color{blue}{-y} \]

   5. Simplified 74.0%

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

Alternative 13: 57.5% accurate, 35.8× speedup

Math

\[\begin{array}{l} \\ \left(-y\right) - z \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (- y) z))

C

double code(double x, double y, double z, double t) {
	return -y - z;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = -y - z
end function

Java

public static double code(double x, double y, double z, double t) {
	return -y - z;
}

Python

def code(x, y, z, t):
	return -y - z

Julia

function code(x, y, z, t)
	return Float64(Float64(-y) - z)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = -y - z;
end

Wolfram

code[x_, y_, z_, t_] := N[((-y) - z), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
2. Add Preprocessing
3. Step-by-step derivation

   1. flip3-+ N/A

      \[\leadsto \color{blue}{\frac{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}} \]

   2. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}}} \]

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

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}}} \]

   4. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}}}} \]

   5. flip3-+ N/A

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in x around inf

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

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

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

   2. log-lowering-log.f64 86.6

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

7. Simplified 86.6%

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

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{-1 \cdot \left(y + z\right)} \]
9. Step-by-step derivation

   1. distribute-lft-in N/A

      \[\leadsto \color{blue}{-1 \cdot y + -1 \cdot z} \]

   2. mul-1-neg N/A

      \[\leadsto -1 \cdot y + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \]

   3. sub-neg N/A

      \[\leadsto \color{blue}{-1 \cdot y - z} \]

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

      \[\leadsto \color{blue}{-1 \cdot y - z} \]

   5. mul-1-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} - z \]

   6. neg-lowering-neg.f64 57.7

      \[\leadsto \color{blue}{\left(-y\right)} - z \]

10. Simplified 57.7%

    \[\leadsto \color{blue}{\left(-y\right) - z} \]
11. Add Preprocessing

Alternative 14: 30.1% accurate, 71.7× speedup

Math

\[\begin{array}{l} \\ -y \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- y))

C

double code(double x, double y, double z, double t) {
	return -y;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = -y
end function

Java

public static double code(double x, double y, double z, double t) {
	return -y;
}

Python

def code(x, y, z, t):
	return -y

Julia

function code(x, y, z, t)
	return Float64(-y)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = -y;
end

Wolfram

code[x_, y_, z_, t_] := (-y)

Derivation

1. Initial program 99.8%

   \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{-1 \cdot y} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(y\right)} \]

   2. neg-lowering-neg.f64 29.6

      \[\leadsto \color{blue}{-y} \]

5. Simplified 29.6%

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

Alternative 15: 2.2% accurate, 215.0× speedup

Math

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

FPCore

(FPCore (x y z t) :precision binary64 y)

C

double code(double x, double y, double z, double t) {
	return y;
}

Fortran

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

Java

public static double code(double x, double y, double z, double t) {
	return y;
}

Python

def code(x, y, z, t):
	return y

Julia

function code(x, y, z, t)
	return y
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = y;
end

Wolfram

code[x_, y_, z_, t_] := y

Derivation

1. Initial program 99.8%

   \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{-1 \cdot y} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(y\right)} \]

   2. neg-lowering-neg.f64 29.6

      \[\leadsto \color{blue}{-y} \]

5. Simplified 29.6%

   \[\leadsto \color{blue}{-y} \]
6. Step-by-step derivation

   1. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - y} \]

   2. sub-neg N/A

      \[\leadsto \color{blue}{0 + \left(\mathsf{neg}\left(y\right)\right)} \]

   3. flip3-+ N/A

      \[\leadsto \color{blue}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(y\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}} \]

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

      \[\leadsto \color{blue}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(y\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0} + {\left(\mathsf{neg}\left(y\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)} \]

   6. sqr-pow N/A

      \[\leadsto \frac{0 + \color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(y\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)} \]

   7. pow-prod-down N/A

      \[\leadsto \frac{0 + \color{blue}{{\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)} \]

   8. sqr-neg N/A

      \[\leadsto \frac{0 + {\color{blue}{\left(y \cdot y\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)} \]

   9. pow-prod-down N/A

      \[\leadsto \frac{0 + \color{blue}{{y}^{\left(\frac{3}{2}\right)} \cdot {y}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)} \]

   10. sqr-pow N/A

       \[\leadsto \frac{0 + \color{blue}{{y}^{3}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)} \]

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

       \[\leadsto \frac{\color{blue}{0 + {y}^{3}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)} \]

   12. cube-mult N/A

       \[\leadsto \frac{0 + \color{blue}{y \cdot \left(y \cdot y\right)}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)} \]

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

       \[\leadsto \frac{0 + \color{blue}{y \cdot \left(y \cdot y\right)}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)} \]

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

       \[\leadsto \frac{0 + y \cdot \color{blue}{\left(y \cdot y\right)}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)} \]

   15. metadata-eval N/A

       \[\leadsto \frac{0 + y \cdot \left(y \cdot y\right)}{\color{blue}{0} + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)} \]

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

       \[\leadsto \frac{0 + y \cdot \left(y \cdot y\right)}{\color{blue}{0 + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}} \]

   17. sqr-neg N/A

       \[\leadsto \frac{0 + y \cdot \left(y \cdot y\right)}{0 + \left(\color{blue}{y \cdot y} - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)} \]

   18. --lowering--.f64 N/A

       \[\leadsto \frac{0 + y \cdot \left(y \cdot y\right)}{0 + \color{blue}{\left(y \cdot y - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}} \]

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

       \[\leadsto \frac{0 + y \cdot \left(y \cdot y\right)}{0 + \left(\color{blue}{y \cdot y} - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)} \]

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

       \[\leadsto \frac{0 + y \cdot \left(y \cdot y\right)}{0 + \left(y \cdot y - \color{blue}{0 \cdot \left(\mathsf{neg}\left(y\right)\right)}\right)} \]

   21. neg-lowering-neg.f64 1.5

       \[\leadsto \frac{0 + y \cdot \left(y \cdot y\right)}{0 + \left(y \cdot y - 0 \cdot \color{blue}{\left(-y\right)}\right)} \]

7. Applied egg-rr 1.5%

   \[\leadsto \color{blue}{\frac{0 + y \cdot \left(y \cdot y\right)}{0 + \left(y \cdot y - 0 \cdot \left(-y\right)\right)}} \]
8. Step-by-step derivation

   1. +-lft-identity N/A

      \[\leadsto \frac{\color{blue}{y \cdot \left(y \cdot y\right)}}{0 + \left(y \cdot y - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)} \]

   2. cube-unmult N/A

      \[\leadsto \frac{\color{blue}{{y}^{3}}}{0 + \left(y \cdot y - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)} \]

   3. +-lft-identity N/A

      \[\leadsto \frac{{y}^{3}}{\color{blue}{y \cdot y - 0 \cdot \left(\mathsf{neg}\left(y\right)\right)}} \]

   4. mul0-lft N/A

      \[\leadsto \frac{{y}^{3}}{y \cdot y - \color{blue}{0}} \]

   5. --rgt-identity N/A

      \[\leadsto \frac{{y}^{3}}{\color{blue}{y \cdot y}} \]

   6. pow2 N/A

      \[\leadsto \frac{{y}^{3}}{\color{blue}{{y}^{2}}} \]

   7. pow-div N/A

      \[\leadsto \color{blue}{{y}^{\left(3 - 2\right)}} \]

   8. metadata-eval N/A

      \[\leadsto {y}^{\color{blue}{1}} \]

   9. unpow1 2.3

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

9. Applied egg-rr 2.3%

   \[\leadsto \color{blue}{y} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024205 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (- (* x (log y)) y) z) (log t)))