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

Percentage Accurate: 99.9% → 99.9%

Time: 11.2s

Alternatives: 12

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, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \log y, \log t - z\right) - y \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-+l- 99.8%

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

   2. sub-neg 99.8%

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

   3. sub-neg 99.8%

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

   4. +-commutative 99.8%

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

   5. associate-+l+ 99.8%

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

   6. +-commutative 99.8%

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

   7. unsub-neg 99.8%

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

   8. fma-udef 99.8%

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

   9. neg-sub0 99.8%

      \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{0 - \left(z - \log t\right)}\right) - y \]

   10. associate-+l- 99.8%

       \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{\left(0 - z\right) + \log t}\right) - y \]

   11. neg-sub0 99.8%

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

   12. +-commutative 99.8%

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

   13. unsub-neg 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

   \[\leadsto \mathsf{fma}\left(x, \log y, \log t - z\right) - y \]
6. Add Preprocessing

Alternative 2: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \left(t\_1 - y\right) - z\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -y\right) - z\\ \mathbf{elif}\;t\_2 \leq 10^{+16}:\\ \;\;\;\;\left(\log t + t\_1\right) - y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- (- t_1 y) z)))
   (if (<= t_2 -1e+18)
     (- (fma (log y) x (- y)) z)
     (if (<= t_2 1e+16) (- (+ (log t) t_1) y) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = (t_1 - y) - z;
	double tmp;
	if (t_2 <= -1e+18) {
		tmp = fma(log(y), x, -y) - z;
	} else if (t_2 <= 1e+16) {
		tmp = (log(t) + t_1) - y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(Float64(t_1 - y) - z)
	tmp = 0.0
	if (t_2 <= -1e+18)
		tmp = Float64(fma(log(y), x, Float64(-y)) - z);
	elseif (t_2 <= 1e+16)
		tmp = Float64(Float64(log(t) + t_1) - y);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 - y), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+18], N[(N[(N[Log[y], $MachinePrecision] * x + (-y)), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[t$95$2, 1e+16], N[(N[(N[Log[t], $MachinePrecision] + t$95$1), $MachinePrecision] - y), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) < -1e18

   1. Initial program 99.8%

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

      1. associate-+l- 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 99.8%

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

      1. add-cbrt-cube 66.6%

         \[\leadsto \left(\color{blue}{\sqrt[3]{\left(\left(x \cdot \log y\right) \cdot \left(x \cdot \log y\right)\right) \cdot \left(x \cdot \log y\right)}} - y\right) - z \]

      2. pow3 66.6%

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

   7. Applied egg-rr 66.6%

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

   8. Taylor expanded in x around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. neg-mul-1 99.8%

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

      3. +-commutative 99.8%

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

      4. fma-def 99.8%

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

   10. Simplified 99.8%

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

   if -1e18 < (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) < 1e16

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. associate--l+ 99.9%

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

      3. fma-def 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 99.9%

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

   if 1e16 < (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z)

   1. Initial program 99.8%

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

      1. associate-+l- 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \log y - y\right) - z \leq -1 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -y\right) - z\\ \mathbf{elif}\;\left(x \cdot \log y - y\right) - z \leq 10^{+16}:\\ \;\;\;\;\left(\log t + x \cdot \log y\right) - y\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log y - y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \left(t\_1 - y\right) - z\\ \mathbf{if}\;t\_2 \leq -20000000 \lor \neg \left(t\_2 \leq 10^{+16}\right):\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\log t + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- (- t_1 y) z)))
   (if (or (<= t_2 -20000000.0) (not (<= t_2 1e+16))) t_2 (+ (log t) 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) - z;
	double tmp;
	if ((t_2 <= -20000000.0) || !(t_2 <= 1e+16)) {
		tmp = t_2;
	} else {
		tmp = log(t) + 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) - z
    if ((t_2 <= (-20000000.0d0)) .or. (.not. (t_2 <= 1d+16))) then
        tmp = t_2
    else
        tmp = log(t) + 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) - z;
	double tmp;
	if ((t_2 <= -20000000.0) || !(t_2 <= 1e+16)) {
		tmp = t_2;
	} else {
		tmp = Math.log(t) + t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = (t_1 - y) - z
	tmp = 0
	if (t_2 <= -20000000.0) or not (t_2 <= 1e+16):
		tmp = t_2
	else:
		tmp = math.log(t) + t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(Float64(t_1 - y) - z)
	tmp = 0.0
	if ((t_2 <= -20000000.0) || !(t_2 <= 1e+16))
		tmp = t_2;
	else
		tmp = Float64(log(t) + 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) - z;
	tmp = 0.0;
	if ((t_2 <= -20000000.0) || ~((t_2 <= 1e+16)))
		tmp = t_2;
	else
		tmp = log(t) + 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[(N[(t$95$1 - y), $MachinePrecision] - z), $MachinePrecision]}, If[Or[LessEqual[t$95$2, -20000000.0], N[Not[LessEqual[t$95$2, 1e+16]], $MachinePrecision]], t$95$2, N[(N[Log[t], $MachinePrecision] + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) < -2e7 or 1e16 < (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z)

   1. Initial program 99.8%

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

      1. associate-+l- 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 99.3%

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

   if -2e7 < (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) < 1e16

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. associate--l+ 99.9%

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

      3. fma-def 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 98.9%

      \[\leadsto \color{blue}{x \cdot \log y} + \log t \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \log y - y\right) - z \leq -20000000 \lor \neg \left(\left(x \cdot \log y - y\right) - z \leq 10^{+16}\right):\\ \;\;\;\;\left(x \cdot \log y - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\log t + x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \left(t\_1 - y\right) - z\\ \mathbf{if}\;t\_2 \leq -20000000:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -y\right) - z\\ \mathbf{elif}\;t\_2 \leq 10^{+16}:\\ \;\;\;\;\log t + t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- (- t_1 y) z)))
   (if (<= t_2 -20000000.0)
     (- (fma (log y) x (- y)) z)
     (if (<= t_2 1e+16) (+ (log t) t_1) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = (t_1 - y) - z;
	double tmp;
	if (t_2 <= -20000000.0) {
		tmp = fma(log(y), x, -y) - z;
	} else if (t_2 <= 1e+16) {
		tmp = log(t) + t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(Float64(t_1 - y) - z)
	tmp = 0.0
	if (t_2 <= -20000000.0)
		tmp = Float64(fma(log(y), x, Float64(-y)) - z);
	elseif (t_2 <= 1e+16)
		tmp = Float64(log(t) + t_1);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 - y), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$2, -20000000.0], N[(N[(N[Log[y], $MachinePrecision] * x + (-y)), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[t$95$2, 1e+16], N[(N[Log[t], $MachinePrecision] + t$95$1), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) < -2e7

   1. Initial program 99.8%

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

      1. associate-+l- 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 99.1%

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

      1. add-cbrt-cube 66.8%

         \[\leadsto \left(\color{blue}{\sqrt[3]{\left(\left(x \cdot \log y\right) \cdot \left(x \cdot \log y\right)\right) \cdot \left(x \cdot \log y\right)}} - y\right) - z \]

      2. pow3 66.8%

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

   7. Applied egg-rr 66.8%

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

   8. Taylor expanded in x around 0 99.1%

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

      1. *-commutative 99.1%

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

      2. neg-mul-1 99.1%

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

      3. +-commutative 99.1%

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

      4. fma-def 99.1%

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

   10. Simplified 99.1%

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

   if -2e7 < (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) < 1e16

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. associate--l+ 99.9%

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

      3. fma-def 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 98.9%

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

   if 1e16 < (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z)

   1. Initial program 99.8%

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

      1. associate-+l- 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 99.8%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \log y - y\right) - z \leq -20000000:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -y\right) - z\\ \mathbf{elif}\;\left(x \cdot \log y - y\right) - z \leq 10^{+16}:\\ \;\;\;\;\log t + x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log y - y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \log y - y\right) - z\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+25} \lor \neg \left(t\_1 \leq 500\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (- (* x (log y)) y) z)))
   (if (or (<= t_1 -1e+25) (not (<= t_1 500.0))) t_1 (- (- (log t) z) y))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((x * log(y)) - y) - z;
	double tmp;
	if ((t_1 <= -1e+25) || !(t_1 <= 500.0)) {
		tmp = t_1;
	} else {
		tmp = (log(t) - z) - 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) :: t_1
    real(8) :: tmp
    t_1 = ((x * log(y)) - y) - z
    if ((t_1 <= (-1d+25)) .or. (.not. (t_1 <= 500.0d0))) then
        tmp = t_1
    else
        tmp = (log(t) - z) - y
    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 <= -1e+25) || !(t_1 <= 500.0)) {
		tmp = t_1;
	} else {
		tmp = (Math.log(t) - z) - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((x * math.log(y)) - y) - z
	tmp = 0
	if (t_1 <= -1e+25) or not (t_1 <= 500.0):
		tmp = t_1
	else:
		tmp = (math.log(t) - z) - y
	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 <= -1e+25) || !(t_1 <= 500.0))
		tmp = t_1;
	else
		tmp = Float64(Float64(log(t) - z) - y);
	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 <= -1e+25) || ~((t_1 <= 500.0)))
		tmp = t_1;
	else
		tmp = (log(t) - z) - y;
	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[Or[LessEqual[t$95$1, -1e+25], N[Not[LessEqual[t$95$1, 500.0]], $MachinePrecision]], t$95$1, N[(N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision] - y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) < -1.00000000000000009e25 or 500 < (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z)

   1. Initial program 99.8%

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

      1. associate-+l- 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 99.1%

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

   if -1.00000000000000009e25 < (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) < 500

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. sub-neg 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+l+ 100.0%

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

      6. +-commutative 100.0%

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

      7. unsub-neg 100.0%

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

      8. fma-udef 100.0%

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

      9. neg-sub0 100.0%

         \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{0 - \left(z - \log t\right)}\right) - y \]

      10. associate-+l- 100.0%

          \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{\left(0 - z\right) + \log t}\right) - y \]

      11. neg-sub0 100.0%

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

      12. +-commutative 100.0%

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

      13. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 88.7%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \log y - y\right) - z \leq -1 \cdot 10^{+25} \lor \neg \left(\left(x \cdot \log y - y\right) - z \leq 500\right):\\ \;\;\;\;\left(x \cdot \log y - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return log(t) + (((x * log(y)) - 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 = log(t) + (((x * log(y)) - y) - z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 7: 69.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+14} \lor \neg \left(z \leq 1.1 \cdot 10^{-6}\right):\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\log t - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.2e+14) (not (<= z 1.1e-6))) (- (- y) z) (- (log t) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.2e+14) || !(z <= 1.1e-6)) {
		tmp = -y - z;
	} else {
		tmp = log(t) - 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 ((z <= (-1.2d+14)) .or. (.not. (z <= 1.1d-6))) then
        tmp = -y - z
    else
        tmp = log(t) - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.2e+14) || !(z <= 1.1e-6)) {
		tmp = -y - z;
	} else {
		tmp = Math.log(t) - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.2e+14) or not (z <= 1.1e-6):
		tmp = -y - z
	else:
		tmp = math.log(t) - y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.2e+14) || !(z <= 1.1e-6))
		tmp = Float64(Float64(-y) - z);
	else
		tmp = Float64(log(t) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.2e+14) || ~((z <= 1.1e-6)))
		tmp = -y - z;
	else
		tmp = log(t) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.2e+14], N[Not[LessEqual[z, 1.1e-6]], $MachinePrecision]], N[((-y) - z), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2e14 or 1.1000000000000001e-6 < z

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 99.9%

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

      1. add-cbrt-cube 70.8%

         \[\leadsto \left(\color{blue}{\sqrt[3]{\left(\left(x \cdot \log y\right) \cdot \left(x \cdot \log y\right)\right) \cdot \left(x \cdot \log y\right)}} - y\right) - z \]

      2. pow3 70.8%

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

   7. Applied egg-rr 70.8%

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

   8. Taylor expanded in x around 0 79.9%

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

      1. neg-mul-1 79.9%

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

   10. Simplified 79.9%

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

   if -1.2e14 < z < 1.1000000000000001e-6

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. associate--l+ 99.7%

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

      3. fma-def 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 54.5%

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

      1. mul-1-neg 54.5%

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

   7. Simplified 54.5%

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

   8. Taylor expanded in y around 0 54.5%

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

      1. mul-1-neg 54.5%

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

      2. sub-neg 54.5%

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

   10. Simplified 54.5%

       \[\leadsto \color{blue}{\log t - y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+14} \lor \neg \left(z \leq 1.1 \cdot 10^{-6}\right):\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\log t - y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.00165:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 0.00165) (- (log t) z) (- (- y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 0.00165) {
		tmp = log(t) - z;
	} 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) :: tmp
    if (y <= 0.00165d0) then
        tmp = log(t) - z
    else
        tmp = -y - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 0.00165) {
		tmp = Math.log(t) - z;
	} else {
		tmp = -y - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 0.00165:
		tmp = math.log(t) - z
	else:
		tmp = -y - z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 0.00165)
		tmp = Float64(log(t) - z);
	else
		tmp = Float64(Float64(-y) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 0.00165)
		tmp = log(t) - z;
	else
		tmp = -y - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 0.00165], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], N[((-y) - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.00165

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. associate--l+ 99.8%

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

      3. fma-def 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. fma-def 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around 0 56.8%

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

   if 0.00165 < y

   1. Initial program 99.8%

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

      1. associate-+l- 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 99.0%

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

      1. add-cbrt-cube 66.0%

         \[\leadsto \left(\color{blue}{\sqrt[3]{\left(\left(x \cdot \log y\right) \cdot \left(x \cdot \log y\right)\right) \cdot \left(x \cdot \log y\right)}} - y\right) - z \]

      2. pow3 66.0%

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

   7. Applied egg-rr 66.0%

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

   8. Taylor expanded in x around 0 75.9%

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

      1. neg-mul-1 75.9%

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

   10. Simplified 75.9%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.00165:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (log(t) - z) - 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 = (log(t) - z) - y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-+l- 99.8%

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

   2. sub-neg 99.8%

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

   3. sub-neg 99.8%

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

   4. +-commutative 99.8%

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

   5. associate-+l+ 99.8%

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

   6. +-commutative 99.8%

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

   7. unsub-neg 99.8%

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

   8. fma-udef 99.8%

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

   9. neg-sub0 99.8%

      \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{0 - \left(z - \log t\right)}\right) - y \]

   10. associate-+l- 99.8%

       \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{\left(0 - z\right) + \log t}\right) - y \]

   11. neg-sub0 99.8%

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

   12. +-commutative 99.8%

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

   13. unsub-neg 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 66.6%

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

6. Final simplification 66.6%

   \[\leadsto \left(\log t - z\right) - y \]
7. Add Preprocessing

Alternative 10: 48.5% accurate, 29.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{+76}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 2.6e+76) (- z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.6e+76) {
		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 <= 2.6d+76) 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 <= 2.6e+76) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 2.6e+76:
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.6e+76)
		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 <= 2.6e+76)
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 2.6e+76], (-z), (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 2.5999999999999999e76

   1. Initial program 99.8%

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

      1. associate-+l- 99.8%

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

   3. Simplified 99.8%

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

      1. add-cube-cbrt 99.1%

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

      2. pow3 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in z around inf 38.0%

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

      1. neg-mul-1 38.0%

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

   9. Simplified 38.0%

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

   if 2.5999999999999999e76 < y

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

   3. Simplified 99.9%

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

      1. add-cube-cbrt 99.6%

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

      2. pow3 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around inf 68.4%

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

      1. mul-1-neg 68.4%

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

   9. Simplified 68.4%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{+76}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.4% accurate, 52.3× 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. Step-by-step derivation

   1. associate-+l- 99.8%

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

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]
4. Add Preprocessing

5. Taylor expanded in z around inf 88.0%

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

   1. add-cbrt-cube 56.2%

      \[\leadsto \left(\color{blue}{\sqrt[3]{\left(\left(x \cdot \log y\right) \cdot \left(x \cdot \log y\right)\right) \cdot \left(x \cdot \log y\right)}} - y\right) - z \]

   2. pow3 56.2%

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

7. Applied egg-rr 56.2%

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

8. Taylor expanded in x around 0 56.6%

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

   1. neg-mul-1 56.6%

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

10. Simplified 56.6%

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

11. Final simplification 56.6%

    \[\leadsto \left(-y\right) - z \]
12. Add Preprocessing

Alternative 12: 30.5% accurate, 104.5× 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. Step-by-step derivation

   1. associate-+l- 99.8%

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

3. Simplified 99.8%

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

   1. add-cube-cbrt 99.3%

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

   2. pow3 99.3%

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

6. Applied egg-rr 99.3%

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

7. Taylor expanded in y around inf 29.8%

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

   1. mul-1-neg 29.8%

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

9. Simplified 29.8%

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

10. Final simplification 29.8%

    \[\leadsto -y \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024027 
(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)))