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

Percentage Accurate: 99.9% → 99.9%

Time: 9.6s

Alternatives: 10

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative 99.9%

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

   2. associate--l- 99.9%

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

   3. associate-+r- 99.9%

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

   4. +-commutative 99.9%

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

   5. associate--l+ 99.9%

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

   6. fma-def 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.4 \cdot 10^{+55} \lor \neg \left(z \leq 6.8 \cdot 10^{+67}\right):\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log t + x \cdot \log y\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -7.4e+55) (not (<= z 6.8e+67)))
   (- (- (log t) y) z)
   (- (+ (log t) (* x (log y))) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -7.4e+55) || !(z <= 6.8e+67)) {
		tmp = (log(t) - y) - z;
	} else {
		tmp = (log(t) + (x * log(y))) - 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 <= (-7.4d+55)) .or. (.not. (z <= 6.8d+67))) then
        tmp = (log(t) - y) - z
    else
        tmp = (log(t) + (x * log(y))) - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -7.4e+55) || !(z <= 6.8e+67)) {
		tmp = (Math.log(t) - y) - z;
	} else {
		tmp = (Math.log(t) + (x * Math.log(y))) - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -7.4e+55) or not (z <= 6.8e+67):
		tmp = (math.log(t) - y) - z
	else:
		tmp = (math.log(t) + (x * math.log(y))) - y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -7.4e+55) || !(z <= 6.8e+67))
		tmp = Float64(Float64(log(t) - y) - z);
	else
		tmp = Float64(Float64(log(t) + Float64(x * log(y))) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -7.4e+55) || ~((z <= 6.8e+67)))
		tmp = (log(t) - y) - z;
	else
		tmp = (log(t) + (x * log(y))) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -7.4e+55], N[Not[LessEqual[z, 6.8e+67]], $MachinePrecision]], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.4000000000000004e55 or 6.8000000000000003e67 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 87.4%

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

      1. associate--r+ 87.4%

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

   4. Simplified 87.4%

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

   if -7.4000000000000004e55 < z < 6.8000000000000003e67

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 97.5%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.4 \cdot 10^{+55} \lor \neg \left(z \leq 6.8 \cdot 10^{+67}\right):\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log t + x \cdot \log y\right) - y\\ \end{array} \]

Alternative 3: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t + x \cdot \log y\\ \mathbf{if}\;z \leq -3.4 \cdot 10^{+61}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+37}:\\ \;\;\;\;t_1 - y\\ \mathbf{else}:\\ \;\;\;\;t_1 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (log t) (* x (log y)))))
   (if (<= z -3.4e+61)
     (- (- (log t) y) z)
     (if (<= z 4.2e+37) (- t_1 y) (- t_1 z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(t) + (x * log(y));
	double tmp;
	if (z <= -3.4e+61) {
		tmp = (log(t) - y) - z;
	} else if (z <= 4.2e+37) {
		tmp = t_1 - y;
	} 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) :: tmp
    t_1 = log(t) + (x * log(y))
    if (z <= (-3.4d+61)) then
        tmp = (log(t) - y) - z
    else if (z <= 4.2d+37) then
        tmp = t_1 - y
    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 = Math.log(t) + (x * Math.log(y));
	double tmp;
	if (z <= -3.4e+61) {
		tmp = (Math.log(t) - y) - z;
	} else if (z <= 4.2e+37) {
		tmp = t_1 - y;
	} else {
		tmp = t_1 - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(t) + (x * math.log(y))
	tmp = 0
	if z <= -3.4e+61:
		tmp = (math.log(t) - y) - z
	elif z <= 4.2e+37:
		tmp = t_1 - y
	else:
		tmp = t_1 - z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(t) + Float64(x * log(y)))
	tmp = 0.0
	if (z <= -3.4e+61)
		tmp = Float64(Float64(log(t) - y) - z);
	elseif (z <= 4.2e+37)
		tmp = Float64(t_1 - y);
	else
		tmp = Float64(t_1 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(t) + (x * log(y));
	tmp = 0.0;
	if (z <= -3.4e+61)
		tmp = (log(t) - y) - z;
	elseif (z <= 4.2e+37)
		tmp = t_1 - y;
	else
		tmp = t_1 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.4e+61], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[z, 4.2e+37], N[(t$95$1 - y), $MachinePrecision], N[(t$95$1 - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.40000000000000026e61

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 89.8%

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

      1. associate--r+ 89.8%

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

   4. Simplified 89.8%

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

   if -3.40000000000000026e61 < z < 4.2000000000000002e37

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 98.0%

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

   if 4.2000000000000002e37 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 86.1%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+61}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+37}:\\ \;\;\;\;\left(\log t + x \cdot \log y\right) - y\\ \mathbf{else}:\\ \;\;\;\;\left(\log t + x \cdot \log y\right) - z\\ \end{array} \]

Alternative 4: 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 99.9%

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

2. Final simplification 99.9%

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

Alternative 5: 60.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t - y\\ \mathbf{if}\;z \leq -2 \cdot 10^{+75}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-242}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-192}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+38}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (log t) y)))
   (if (<= z -2e+75)
     (- z)
     (if (<= z 1.3e-242)
       t_1
       (if (<= z 6.5e-192) (* x (log y)) (if (<= z 5.8e+38) t_1 (- z)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(t) - y;
	double tmp;
	if (z <= -2e+75) {
		tmp = -z;
	} else if (z <= 1.3e-242) {
		tmp = t_1;
	} else if (z <= 6.5e-192) {
		tmp = x * log(y);
	} else if (z <= 5.8e+38) {
		tmp = t_1;
	} else {
		tmp = -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 = log(t) - y
    if (z <= (-2d+75)) then
        tmp = -z
    else if (z <= 1.3d-242) then
        tmp = t_1
    else if (z <= 6.5d-192) then
        tmp = x * log(y)
    else if (z <= 5.8d+38) then
        tmp = t_1
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(t) - y;
	double tmp;
	if (z <= -2e+75) {
		tmp = -z;
	} else if (z <= 1.3e-242) {
		tmp = t_1;
	} else if (z <= 6.5e-192) {
		tmp = x * Math.log(y);
	} else if (z <= 5.8e+38) {
		tmp = t_1;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(t) - y
	tmp = 0
	if z <= -2e+75:
		tmp = -z
	elif z <= 1.3e-242:
		tmp = t_1
	elif z <= 6.5e-192:
		tmp = x * math.log(y)
	elif z <= 5.8e+38:
		tmp = t_1
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(t) - y)
	tmp = 0.0
	if (z <= -2e+75)
		tmp = Float64(-z);
	elseif (z <= 1.3e-242)
		tmp = t_1;
	elseif (z <= 6.5e-192)
		tmp = Float64(x * log(y));
	elseif (z <= 5.8e+38)
		tmp = t_1;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(t) - y;
	tmp = 0.0;
	if (z <= -2e+75)
		tmp = -z;
	elseif (z <= 1.3e-242)
		tmp = t_1;
	elseif (z <= 6.5e-192)
		tmp = x * log(y);
	elseif (z <= 5.8e+38)
		tmp = t_1;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[z, -2e+75], (-z), If[LessEqual[z, 1.3e-242], t$95$1, If[LessEqual[z, 6.5e-192], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.8e+38], t$95$1, (-z)]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.99999999999999985e75 or 5.80000000000000013e38 < 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%

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

      2. associate-+l- 99.9%

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

      3. *-commutative 99.9%

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

      4. add-cube-cbrt 99.8%

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

      5. associate-*l* 99.8%

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

      6. fma-neg 99.8%

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

      7. pow2 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. sub-neg 99.8%

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

      2. log-rec 99.8%

         \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, \sqrt[3]{\log y} \cdot x, -\left(\left(y + z\right) + \color{blue}{\log \left(\frac{1}{t}\right)}\right)\right) \]

      3. distribute-neg-in 99.8%

         \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, \sqrt[3]{\log y} \cdot x, \color{blue}{\left(-\left(y + z\right)\right) + \left(-\log \left(\frac{1}{t}\right)\right)}\right) \]

      4. log-rec 99.8%

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

      5. remove-double-neg 99.8%

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

      6. +-commutative 99.8%

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

      7. sub-neg 99.8%

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

      8. *-commutative 99.8%

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

      9. associate--r+ 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in z around inf 72.3%

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

      1. neg-mul-1 72.3%

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

   8. Simplified 72.3%

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

   if -1.99999999999999985e75 < z < 1.30000000000000009e-242 or 6.49999999999999966e-192 < z < 5.80000000000000013e38

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 96.5%

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

   3. Taylor expanded in x around 0 67.0%

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

   if 1.30000000000000009e-242 < z < 6.49999999999999966e-192

   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%

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

      2. associate-+l- 99.9%

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

      3. *-commutative 99.9%

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

      4. add-cube-cbrt 98.8%

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

      5. associate-*l* 99.0%

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

      6. fma-neg 99.0%

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

      7. pow2 99.0%

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

   3. Applied egg-rr 99.0%

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

      1. sub-neg 99.0%

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

      2. log-rec 99.0%

         \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, \sqrt[3]{\log y} \cdot x, -\left(\left(y + z\right) + \color{blue}{\log \left(\frac{1}{t}\right)}\right)\right) \]

      3. distribute-neg-in 99.0%

         \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, \sqrt[3]{\log y} \cdot x, \color{blue}{\left(-\left(y + z\right)\right) + \left(-\log \left(\frac{1}{t}\right)\right)}\right) \]

      4. log-rec 99.0%

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

      5. remove-double-neg 99.0%

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

      6. +-commutative 99.0%

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

      7. sub-neg 99.0%

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

      8. *-commutative 99.0%

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

      9. associate--r+ 99.0%

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

   5. Simplified 99.0%

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

   6. Taylor expanded in x around inf 67.6%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \left(\log y \cdot x\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 67.6%

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

      2. *-lft-identity 67.6%

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

   8. Simplified 67.6%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+75}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-242}:\\ \;\;\;\;\log t - y\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-192}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+38}:\\ \;\;\;\;\log t - y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 6: 59.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t - z\\ t_2 := x \cdot \log y\\ \mathbf{if}\;y \leq 3.2 \cdot 10^{-279}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-275}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+133}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (log t) z)) (t_2 (* x (log y))))
   (if (<= y 3.2e-279)
     t_1
     (if (<= y 6.5e-275)
       t_2
       (if (<= y 6.2e+109) t_1 (if (<= y 2.6e+133) t_2 (- y)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(t) - z;
	double t_2 = x * log(y);
	double tmp;
	if (y <= 3.2e-279) {
		tmp = t_1;
	} else if (y <= 6.5e-275) {
		tmp = t_2;
	} else if (y <= 6.2e+109) {
		tmp = t_1;
	} else if (y <= 2.6e+133) {
		tmp = t_2;
	} 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = log(t) - z
    t_2 = x * log(y)
    if (y <= 3.2d-279) then
        tmp = t_1
    else if (y <= 6.5d-275) then
        tmp = t_2
    else if (y <= 6.2d+109) then
        tmp = t_1
    else if (y <= 2.6d+133) then
        tmp = t_2
    else
        tmp = -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(t) - z;
	double t_2 = x * Math.log(y);
	double tmp;
	if (y <= 3.2e-279) {
		tmp = t_1;
	} else if (y <= 6.5e-275) {
		tmp = t_2;
	} else if (y <= 6.2e+109) {
		tmp = t_1;
	} else if (y <= 2.6e+133) {
		tmp = t_2;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(t) - z
	t_2 = x * math.log(y)
	tmp = 0
	if y <= 3.2e-279:
		tmp = t_1
	elif y <= 6.5e-275:
		tmp = t_2
	elif y <= 6.2e+109:
		tmp = t_1
	elif y <= 2.6e+133:
		tmp = t_2
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(t) - z)
	t_2 = Float64(x * log(y))
	tmp = 0.0
	if (y <= 3.2e-279)
		tmp = t_1;
	elseif (y <= 6.5e-275)
		tmp = t_2;
	elseif (y <= 6.2e+109)
		tmp = t_1;
	elseif (y <= 2.6e+133)
		tmp = t_2;
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(t) - z;
	t_2 = x * log(y);
	tmp = 0.0;
	if (y <= 3.2e-279)
		tmp = t_1;
	elseif (y <= 6.5e-275)
		tmp = t_2;
	elseif (y <= 6.2e+109)
		tmp = t_1;
	elseif (y <= 2.6e+133)
		tmp = t_2;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 3.2e-279], t$95$1, If[LessEqual[y, 6.5e-275], t$95$2, If[LessEqual[y, 6.2e+109], t$95$1, If[LessEqual[y, 2.6e+133], t$95$2, (-y)]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.1999999999999999e-279 or 6.500000000000001e-275 < y < 6.19999999999999985e109

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 92.4%

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

      1. associate--l+ 92.4%

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

      2. fma-def 92.4%

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

   4. Simplified 92.4%

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

   5. Taylor expanded in x around 0 66.0%

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

   if 3.1999999999999999e-279 < y < 6.500000000000001e-275 or 6.19999999999999985e109 < y < 2.5999999999999998e133

   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%

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

      2. associate-+l- 99.7%

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

      3. *-commutative 99.7%

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

      4. add-cube-cbrt 98.6%

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

      5. associate-*l* 98.4%

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

      6. fma-neg 98.4%

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

      7. pow2 98.4%

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

   3. Applied egg-rr 98.4%

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

      1. sub-neg 98.4%

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

      2. log-rec 98.4%

         \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, \sqrt[3]{\log y} \cdot x, -\left(\left(y + z\right) + \color{blue}{\log \left(\frac{1}{t}\right)}\right)\right) \]

      3. distribute-neg-in 98.4%

         \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, \sqrt[3]{\log y} \cdot x, \color{blue}{\left(-\left(y + z\right)\right) + \left(-\log \left(\frac{1}{t}\right)\right)}\right) \]

      4. log-rec 98.4%

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

      5. remove-double-neg 98.4%

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

      6. +-commutative 98.4%

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

      7. sub-neg 98.4%

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

      8. *-commutative 98.4%

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

      9. associate--r+ 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in x around inf 75.8%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \left(\log y \cdot x\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 75.8%

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

      2. *-lft-identity 75.8%

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

   8. Simplified 75.8%

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

   if 2.5999999999999998e133 < y

   1. Initial program 100.0%

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

      1. associate--l- 100.0%

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

      2. associate-+l- 100.0%

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

      3. *-commutative 100.0%

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

      4. add-cube-cbrt 99.8%

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

      5. associate-*l* 99.8%

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

      6. fma-neg 99.8%

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

      7. pow2 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. sub-neg 99.8%

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

      2. log-rec 99.8%

         \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, \sqrt[3]{\log y} \cdot x, -\left(\left(y + z\right) + \color{blue}{\log \left(\frac{1}{t}\right)}\right)\right) \]

      3. distribute-neg-in 99.8%

         \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, \sqrt[3]{\log y} \cdot x, \color{blue}{\left(-\left(y + z\right)\right) + \left(-\log \left(\frac{1}{t}\right)\right)}\right) \]

      4. log-rec 99.8%

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

      5. remove-double-neg 99.8%

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

      6. +-commutative 99.8%

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

      7. sub-neg 99.8%

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

      8. *-commutative 99.8%

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

      9. associate--r+ 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around inf 82.4%

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

      1. neg-mul-1 82.4%

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

   8. Simplified 82.4%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-279}:\\ \;\;\;\;\log t - z\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-275}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+109}:\\ \;\;\;\;\log t - z\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+133}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 7: 48.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+75}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -1.85 \cdot 10^{-270}:\\ \;\;\;\;-y\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-135}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+38}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.9e+75)
   (- z)
   (if (<= z -1.85e-270)
     (- y)
     (if (<= z 1.3e-135) (* x (log y)) (if (<= z 5.4e+38) (- y) (- z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.9e+75) {
		tmp = -z;
	} else if (z <= -1.85e-270) {
		tmp = -y;
	} else if (z <= 1.3e-135) {
		tmp = x * log(y);
	} else if (z <= 5.4e+38) {
		tmp = -y;
	} else {
		tmp = -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 (z <= (-2.9d+75)) then
        tmp = -z
    else if (z <= (-1.85d-270)) then
        tmp = -y
    else if (z <= 1.3d-135) then
        tmp = x * log(y)
    else if (z <= 5.4d+38) then
        tmp = -y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.9e+75) {
		tmp = -z;
	} else if (z <= -1.85e-270) {
		tmp = -y;
	} else if (z <= 1.3e-135) {
		tmp = x * Math.log(y);
	} else if (z <= 5.4e+38) {
		tmp = -y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -2.9e+75:
		tmp = -z
	elif z <= -1.85e-270:
		tmp = -y
	elif z <= 1.3e-135:
		tmp = x * math.log(y)
	elif z <= 5.4e+38:
		tmp = -y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.9e+75)
		tmp = Float64(-z);
	elseif (z <= -1.85e-270)
		tmp = Float64(-y);
	elseif (z <= 1.3e-135)
		tmp = Float64(x * log(y));
	elseif (z <= 5.4e+38)
		tmp = Float64(-y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -2.9e+75)
		tmp = -z;
	elseif (z <= -1.85e-270)
		tmp = -y;
	elseif (z <= 1.3e-135)
		tmp = x * log(y);
	elseif (z <= 5.4e+38)
		tmp = -y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.9e+75], (-z), If[LessEqual[z, -1.85e-270], (-y), If[LessEqual[z, 1.3e-135], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.4e+38], (-y), (-z)]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.8999999999999998e75 or 5.39999999999999992e38 < 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%

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

      2. associate-+l- 99.9%

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

      3. *-commutative 99.9%

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

      4. add-cube-cbrt 99.8%

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

      5. associate-*l* 99.8%

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

      6. fma-neg 99.8%

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

      7. pow2 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. sub-neg 99.8%

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

      2. log-rec 99.8%

         \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, \sqrt[3]{\log y} \cdot x, -\left(\left(y + z\right) + \color{blue}{\log \left(\frac{1}{t}\right)}\right)\right) \]

      3. distribute-neg-in 99.8%

         \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, \sqrt[3]{\log y} \cdot x, \color{blue}{\left(-\left(y + z\right)\right) + \left(-\log \left(\frac{1}{t}\right)\right)}\right) \]

      4. log-rec 99.8%

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

      5. remove-double-neg 99.8%

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

      6. +-commutative 99.8%

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

      7. sub-neg 99.8%

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

      8. *-commutative 99.8%

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

      9. associate--r+ 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in z around inf 72.3%

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

      1. neg-mul-1 72.3%

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

   8. Simplified 72.3%

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

   if -2.8999999999999998e75 < z < -1.8500000000000001e-270 or 1.30000000000000002e-135 < z < 5.39999999999999992e38

   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%

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

      2. associate-+l- 99.9%

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

      3. *-commutative 99.9%

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

      4. add-cube-cbrt 99.5%

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

      5. associate-*l* 99.5%

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

      6. fma-neg 99.5%

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

      7. pow2 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. sub-neg 99.5%

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

      2. log-rec 99.5%

         \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, \sqrt[3]{\log y} \cdot x, -\left(\left(y + z\right) + \color{blue}{\log \left(\frac{1}{t}\right)}\right)\right) \]

      3. distribute-neg-in 99.5%

         \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, \sqrt[3]{\log y} \cdot x, \color{blue}{\left(-\left(y + z\right)\right) + \left(-\log \left(\frac{1}{t}\right)\right)}\right) \]

      4. log-rec 99.5%

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

      5. remove-double-neg 99.5%

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

      6. +-commutative 99.5%

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

      7. sub-neg 99.5%

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

      8. *-commutative 99.5%

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

      9. associate--r+ 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in y around inf 42.3%

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

      1. neg-mul-1 42.3%

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

   8. Simplified 42.3%

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

   if -1.8500000000000001e-270 < z < 1.30000000000000002e-135

   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%

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

      2. associate-+l- 99.9%

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

      3. *-commutative 99.9%

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

      4. add-cube-cbrt 99.1%

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

      5. associate-*l* 99.0%

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

      6. fma-neg 99.0%

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

      7. pow2 99.0%

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

   3. Applied egg-rr 99.0%

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

      1. sub-neg 99.0%

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

      2. log-rec 99.0%

         \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, \sqrt[3]{\log y} \cdot x, -\left(\left(y + z\right) + \color{blue}{\log \left(\frac{1}{t}\right)}\right)\right) \]

      3. distribute-neg-in 99.0%

         \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, \sqrt[3]{\log y} \cdot x, \color{blue}{\left(-\left(y + z\right)\right) + \left(-\log \left(\frac{1}{t}\right)\right)}\right) \]

      4. log-rec 99.0%

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

      5. remove-double-neg 99.0%

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

      6. +-commutative 99.0%

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

      7. sub-neg 99.0%

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

      8. *-commutative 99.0%

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

      9. associate--r+ 99.0%

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

   5. Simplified 99.0%

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

   6. Taylor expanded in x around inf 52.6%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \left(\log y \cdot x\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 52.6%

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

      2. *-lft-identity 52.6%

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

   8. Simplified 52.6%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+75}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -1.85 \cdot 10^{-270}:\\ \;\;\;\;-y\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-135}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+38}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 8: 84.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+141} \lor \neg \left(x \leq 2.4 \cdot 10^{+149}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -9.5e+141) (not (<= x 2.4e+149)))
   (* x (log y))
   (- (- (log t) y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -9.5e+141) || !(x <= 2.4e+149)) {
		tmp = x * log(y);
	} 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 <= (-9.5d+141)) .or. (.not. (x <= 2.4d+149))) then
        tmp = x * log(y)
    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 <= -9.5e+141) || !(x <= 2.4e+149)) {
		tmp = x * Math.log(y);
	} else {
		tmp = (Math.log(t) - y) - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -9.5e+141) or not (x <= 2.4e+149):
		tmp = x * math.log(y)
	else:
		tmp = (math.log(t) - y) - z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -9.5e+141) || !(x <= 2.4e+149))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(log(t) - y) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -9.5e+141) || ~((x <= 2.4e+149)))
		tmp = x * log(y);
	else
		tmp = (log(t) - y) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -9.5e+141], N[Not[LessEqual[x, 2.4e+149]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.49999999999999974e141 or 2.40000000000000012e149 < 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%

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

      2. associate-+l- 99.7%

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

      3. *-commutative 99.7%

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

      4. add-cube-cbrt 98.5%

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

      5. associate-*l* 98.5%

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

      6. fma-neg 98.5%

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

      7. pow2 98.5%

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

   3. Applied egg-rr 98.5%

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

      1. sub-neg 98.5%

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

      2. log-rec 98.5%

         \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, \sqrt[3]{\log y} \cdot x, -\left(\left(y + z\right) + \color{blue}{\log \left(\frac{1}{t}\right)}\right)\right) \]

      3. distribute-neg-in 98.5%

         \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, \sqrt[3]{\log y} \cdot x, \color{blue}{\left(-\left(y + z\right)\right) + \left(-\log \left(\frac{1}{t}\right)\right)}\right) \]

      4. log-rec 98.5%

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

      5. remove-double-neg 98.5%

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

      6. +-commutative 98.5%

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

      7. sub-neg 98.5%

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

      8. *-commutative 98.5%

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

      9. associate--r+ 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in x around inf 73.2%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \left(\log y \cdot x\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 73.2%

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

      2. *-lft-identity 73.2%

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

   8. Simplified 73.2%

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

   if -9.49999999999999974e141 < x < 2.40000000000000012e149

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 91.2%

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

      1. associate--r+ 91.2%

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

   4. Simplified 91.2%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+141} \lor \neg \left(x \leq 2.4 \cdot 10^{+149}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \end{array} \]

Alternative 9: 48.8% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+75}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{+38}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2e+75) (- z) (if (<= z 6.6e+38) (- y) (- z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2e+75) {
		tmp = -z;
	} else if (z <= 6.6e+38) {
		tmp = -y;
	} else {
		tmp = -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 (z <= (-2d+75)) then
        tmp = -z
    else if (z <= 6.6d+38) then
        tmp = -y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2e+75) {
		tmp = -z;
	} else if (z <= 6.6e+38) {
		tmp = -y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -2e+75:
		tmp = -z
	elif z <= 6.6e+38:
		tmp = -y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2e+75)
		tmp = Float64(-z);
	elseif (z <= 6.6e+38)
		tmp = Float64(-y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -2e+75)
		tmp = -z;
	elseif (z <= 6.6e+38)
		tmp = -y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2e+75], (-z), If[LessEqual[z, 6.6e+38], (-y), (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -1.99999999999999985e75 or 6.5999999999999998e38 < 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%

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

      2. associate-+l- 99.9%

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

      3. *-commutative 99.9%

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

      4. add-cube-cbrt 99.8%

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

      5. associate-*l* 99.8%

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

      6. fma-neg 99.8%

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

      7. pow2 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. sub-neg 99.8%

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

      2. log-rec 99.8%

         \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, \sqrt[3]{\log y} \cdot x, -\left(\left(y + z\right) + \color{blue}{\log \left(\frac{1}{t}\right)}\right)\right) \]

      3. distribute-neg-in 99.8%

         \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, \sqrt[3]{\log y} \cdot x, \color{blue}{\left(-\left(y + z\right)\right) + \left(-\log \left(\frac{1}{t}\right)\right)}\right) \]

      4. log-rec 99.8%

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

      5. remove-double-neg 99.8%

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

      6. +-commutative 99.8%

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

      7. sub-neg 99.8%

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

      8. *-commutative 99.8%

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

      9. associate--r+ 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in z around inf 72.3%

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

      1. neg-mul-1 72.3%

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

   8. Simplified 72.3%

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

   if -1.99999999999999985e75 < z < 6.5999999999999998e38

   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%

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

      2. associate-+l- 99.9%

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

      3. *-commutative 99.9%

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

      4. add-cube-cbrt 99.4%

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

      5. associate-*l* 99.4%

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

      6. fma-neg 99.4%

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

      7. pow2 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. sub-neg 99.4%

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

      2. log-rec 99.4%

         \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, \sqrt[3]{\log y} \cdot x, -\left(\left(y + z\right) + \color{blue}{\log \left(\frac{1}{t}\right)}\right)\right) \]

      3. distribute-neg-in 99.4%

         \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, \sqrt[3]{\log y} \cdot x, \color{blue}{\left(-\left(y + z\right)\right) + \left(-\log \left(\frac{1}{t}\right)\right)}\right) \]

      4. log-rec 99.4%

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

      5. remove-double-neg 99.4%

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

      6. +-commutative 99.4%

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

      7. sub-neg 99.4%

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

      8. *-commutative 99.4%

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

      9. associate--r+ 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in y around inf 37.8%

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

      1. neg-mul-1 37.8%

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

   8. Simplified 37.8%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+75}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{+38}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 10: 30.6% 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 99.9%

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

   1. associate--l- 99.9%

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

   2. associate-+l- 99.9%

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

   3. *-commutative 99.9%

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

   4. add-cube-cbrt 99.5%

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

   5. associate-*l* 99.5%

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

   6. fma-neg 99.5%

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

   7. pow2 99.5%

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

3. Applied egg-rr 99.5%

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

   1. sub-neg 99.5%

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

   2. log-rec 99.5%

      \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, \sqrt[3]{\log y} \cdot x, -\left(\left(y + z\right) + \color{blue}{\log \left(\frac{1}{t}\right)}\right)\right) \]

   3. distribute-neg-in 99.5%

      \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, \sqrt[3]{\log y} \cdot x, \color{blue}{\left(-\left(y + z\right)\right) + \left(-\log \left(\frac{1}{t}\right)\right)}\right) \]

   4. log-rec 99.5%

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

   5. remove-double-neg 99.5%

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

   6. +-commutative 99.5%

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

   7. sub-neg 99.5%

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

   8. *-commutative 99.5%

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

   9. associate--r+ 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in y around inf 29.3%

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

   1. neg-mul-1 29.3%

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

8. Simplified 29.3%

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

9. Final simplification 29.3%

   \[\leadsto -y \]

Reproduce

herbie shell --seed 2023224 
(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)))