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

Percentage Accurate: 99.9% → 99.9%

Time: 10.8s

Alternatives: 7

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. Final simplification 99.8%

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

Alternative 2: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \left(\log t + t_1\right) - y\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{+70}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+42}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{+154}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- (+ (log t) t_1) y)))
   (if (<= x -1.5e+70)
     t_2
     (if (<= x 1.25e+42)
       (- (log t) (+ y z))
       (if (<= x 8.6e+154) t_2 (- t_1 z))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = (log(t) + t_1) - y;
	double tmp;
	if (x <= -1.5e+70) {
		tmp = t_2;
	} else if (x <= 1.25e+42) {
		tmp = log(t) - (y + z);
	} else if (x <= 8.6e+154) {
		tmp = t_2;
	} else {
		tmp = t_1 - 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) :: t_2
    real(8) :: tmp
    t_1 = x * log(y)
    t_2 = (log(t) + t_1) - y
    if (x <= (-1.5d+70)) then
        tmp = t_2
    else if (x <= 1.25d+42) then
        tmp = log(t) - (y + z)
    else if (x <= 8.6d+154) then
        tmp = t_2
    else
        tmp = t_1 - 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);
	double t_2 = (Math.log(t) + t_1) - y;
	double tmp;
	if (x <= -1.5e+70) {
		tmp = t_2;
	} else if (x <= 1.25e+42) {
		tmp = Math.log(t) - (y + z);
	} else if (x <= 8.6e+154) {
		tmp = t_2;
	} else {
		tmp = t_1 - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = (math.log(t) + t_1) - y
	tmp = 0
	if x <= -1.5e+70:
		tmp = t_2
	elif x <= 1.25e+42:
		tmp = math.log(t) - (y + z)
	elif x <= 8.6e+154:
		tmp = t_2
	else:
		tmp = t_1 - z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(Float64(log(t) + t_1) - y)
	tmp = 0.0
	if (x <= -1.5e+70)
		tmp = t_2;
	elseif (x <= 1.25e+42)
		tmp = Float64(log(t) - Float64(y + z));
	elseif (x <= 8.6e+154)
		tmp = t_2;
	else
		tmp = Float64(t_1 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	t_2 = (log(t) + t_1) - y;
	tmp = 0.0;
	if (x <= -1.5e+70)
		tmp = t_2;
	elseif (x <= 1.25e+42)
		tmp = log(t) - (y + z);
	elseif (x <= 8.6e+154)
		tmp = t_2;
	else
		tmp = t_1 - z;
	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[(N[Log[t], $MachinePrecision] + t$95$1), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[x, -1.5e+70], t$95$2, If[LessEqual[x, 1.25e+42], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.6e+154], t$95$2, N[(t$95$1 - z), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.49999999999999988e70 or 1.25000000000000002e42 < x < 8.5999999999999995e154

   1. Initial program 99.7%

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

   2. Taylor expanded in z around 0 88.9%

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

   if -1.49999999999999988e70 < x < 1.25000000000000002e42

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

   if 8.5999999999999995e154 < x

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. sub-neg 99.7%

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

      3. associate--l+ 99.7%

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

      4. add-cube-cbrt 98.5%

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

      5. fma-def 98.5%

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

      6. pow2 98.5%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{x \cdot \log y}\right)}^{2}}, \sqrt[3]{x \cdot \log y}, \left(-y\right) - \left(z - \log t\right)\right) \]

   3. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{x \cdot \log y}\right)}^{2}, \sqrt[3]{x \cdot \log y}, \left(-y\right) - \left(z - \log t\right)\right)} \]

   4. Taylor expanded in z around inf 93.3%

      \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{x \cdot \log y}\right)}^{2}, \sqrt[3]{x \cdot \log y}, \color{blue}{-1 \cdot z}\right) \]
   5. Step-by-step derivation

      1. neg-mul-1 93.3%

         \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{x \cdot \log y}\right)}^{2}, \sqrt[3]{x \cdot \log y}, \color{blue}{-z}\right) \]

   6. Simplified 93.3%

      \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{x \cdot \log y}\right)}^{2}, \sqrt[3]{x \cdot \log y}, \color{blue}{-z}\right) \]
   7. Step-by-step derivation

      1. fma-udef 93.3%

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

      2. unpow2 93.3%

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

      3. add-cube-cbrt 94.4%

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

      4. *-commutative 94.4%

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

      5. unsub-neg 94.4%

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

   8. Applied egg-rr 94.4%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+70}:\\ \;\;\;\;\left(\log t + x \cdot \log y\right) - y\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+42}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{+154}:\\ \;\;\;\;\left(\log t + x \cdot \log y\right) - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - z\\ \end{array} \]

Alternative 3: 88.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+106} \lor \neg \left(x \leq 2.3 \cdot 10^{+96}\right):\\ \;\;\;\;x \cdot \log y - z\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -3.2e+106) (not (<= x 2.3e+96)))
   (- (* x (log y)) z)
   (- (log t) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.2e+106) || !(x <= 2.3e+96)) {
		tmp = (x * log(y)) - z;
	} else {
		tmp = log(t) - (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 ((x <= (-3.2d+106)) .or. (.not. (x <= 2.3d+96))) then
        tmp = (x * log(y)) - z
    else
        tmp = log(t) - (y + z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.2e+106) || !(x <= 2.3e+96)) {
		tmp = (x * Math.log(y)) - z;
	} else {
		tmp = Math.log(t) - (y + z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -3.2e+106) or not (x <= 2.3e+96):
		tmp = (x * math.log(y)) - z
	else:
		tmp = math.log(t) - (y + z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -3.2e+106) || !(x <= 2.3e+96))
		tmp = Float64(Float64(x * log(y)) - z);
	else
		tmp = Float64(log(t) - Float64(y + z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -3.2e+106) || ~((x <= 2.3e+96)))
		tmp = (x * log(y)) - z;
	else
		tmp = log(t) - (y + z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -3.2e+106], N[Not[LessEqual[x, 2.3e+96]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.1999999999999998e106 or 2.30000000000000015e96 < x

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. sub-neg 99.7%

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

      3. associate--l+ 99.7%

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

      4. add-cube-cbrt 98.4%

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

      5. fma-def 98.4%

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

      6. pow2 98.4%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{x \cdot \log y}\right)}^{2}}, \sqrt[3]{x \cdot \log y}, \left(-y\right) - \left(z - \log t\right)\right) \]

   3. Applied egg-rr 98.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{x \cdot \log y}\right)}^{2}, \sqrt[3]{x \cdot \log y}, \left(-y\right) - \left(z - \log t\right)\right)} \]

   4. Taylor expanded in z around inf 87.7%

      \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{x \cdot \log y}\right)}^{2}, \sqrt[3]{x \cdot \log y}, \color{blue}{-1 \cdot z}\right) \]
   5. Step-by-step derivation

      1. neg-mul-1 87.7%

         \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{x \cdot \log y}\right)}^{2}, \sqrt[3]{x \cdot \log y}, \color{blue}{-z}\right) \]

   6. Simplified 87.7%

      \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{x \cdot \log y}\right)}^{2}, \sqrt[3]{x \cdot \log y}, \color{blue}{-z}\right) \]
   7. Step-by-step derivation

      1. fma-udef 87.7%

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

      2. unpow2 87.7%

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

      3. add-cube-cbrt 88.9%

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

      4. *-commutative 88.9%

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

      5. unsub-neg 88.9%

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

   8. Applied egg-rr 88.9%

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

   if -3.1999999999999998e106 < x < 2.30000000000000015e96

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 96.1%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+106} \lor \neg \left(x \leq 2.3 \cdot 10^{+96}\right):\\ \;\;\;\;x \cdot \log y - z\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]

Alternative 4: 60.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{+27}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 1.5e+27) (- (log t) z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.5e+27) {
		tmp = log(t) - 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 <= 1.5d+27) then
        tmp = log(t) - 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 <= 1.5e+27) {
		tmp = Math.log(t) - z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 1.5e+27:
		tmp = math.log(t) - z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.5e+27)
		tmp = Float64(log(t) - z);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 1.5e+27)
		tmp = log(t) - z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.5e+27], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 1.49999999999999988e27

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 97.9%

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

      1. +-commutative 97.9%

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

      2. *-commutative 97.9%

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

      3. log-pow 55.0%

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

      4. log-prod 55.0%

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

   4. Simplified 55.0%

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

   5. Taylor expanded in x around 0 65.7%

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

   if 1.49999999999999988e27 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 59.1%

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

      1. neg-mul-1 59.1%

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

   4. Simplified 59.1%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{+27}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 5: 70.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0 71.3%

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

3. Final simplification 71.3%

   \[\leadsto \log t - \left(y + z\right) \]

Alternative 6: 48.0% accurate, 51.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{+26}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 1.05e+26) (- z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.05e+26) {
		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 <= 1.05d+26) 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 <= 1.05e+26) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 1.05e+26:
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.05e+26)
		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 <= 1.05e+26)
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.05e+26], (-z), (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 1.05e26

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf 47.5%

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

      1. mul-1-neg 47.5%

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

   4. Simplified 47.5%

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

   if 1.05e26 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 59.1%

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

      1. neg-mul-1 59.1%

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

   4. Simplified 59.1%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{+26}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 7: 29.3% 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. Taylor expanded in y around inf 30.6%

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

   1. neg-mul-1 30.6%

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

4. Simplified 30.6%

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

5. Final simplification 30.6%

   \[\leadsto -y \]

Reproduce

herbie shell --seed 2023196 
(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)))