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

Percentage Accurate: 99.9% → 99.9%

Time: 12.3s

Alternatives: 9

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: 58.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \log t - y\\ \mathbf{if}\;x \leq -4.7 \cdot 10^{+67}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-188}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -2.45 \cdot 10^{-256}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-255}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-177}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 9.4 \cdot 10^{+21}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- (log t) y)))
   (if (<= x -4.7e+67)
     t_1
     (if (<= x -1.7e-188)
       t_2
       (if (<= x -2.45e-256)
         (- z)
         (if (<= x 4e-255)
           t_2
           (if (<= x 2.1e-177) (- z) (if (<= x 9.4e+21) t_2 t_1))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = log(t) - y;
	double tmp;
	if (x <= -4.7e+67) {
		tmp = t_1;
	} else if (x <= -1.7e-188) {
		tmp = t_2;
	} else if (x <= -2.45e-256) {
		tmp = -z;
	} else if (x <= 4e-255) {
		tmp = t_2;
	} else if (x <= 2.1e-177) {
		tmp = -z;
	} else if (x <= 9.4e+21) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * log(y)
    t_2 = log(t) - y
    if (x <= (-4.7d+67)) then
        tmp = t_1
    else if (x <= (-1.7d-188)) then
        tmp = t_2
    else if (x <= (-2.45d-256)) then
        tmp = -z
    else if (x <= 4d-255) then
        tmp = t_2
    else if (x <= 2.1d-177) then
        tmp = -z
    else if (x <= 9.4d+21) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double t_2 = Math.log(t) - y;
	double tmp;
	if (x <= -4.7e+67) {
		tmp = t_1;
	} else if (x <= -1.7e-188) {
		tmp = t_2;
	} else if (x <= -2.45e-256) {
		tmp = -z;
	} else if (x <= 4e-255) {
		tmp = t_2;
	} else if (x <= 2.1e-177) {
		tmp = -z;
	} else if (x <= 9.4e+21) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = math.log(t) - y
	tmp = 0
	if x <= -4.7e+67:
		tmp = t_1
	elif x <= -1.7e-188:
		tmp = t_2
	elif x <= -2.45e-256:
		tmp = -z
	elif x <= 4e-255:
		tmp = t_2
	elif x <= 2.1e-177:
		tmp = -z
	elif x <= 9.4e+21:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(log(t) - y)
	tmp = 0.0
	if (x <= -4.7e+67)
		tmp = t_1;
	elseif (x <= -1.7e-188)
		tmp = t_2;
	elseif (x <= -2.45e-256)
		tmp = Float64(-z);
	elseif (x <= 4e-255)
		tmp = t_2;
	elseif (x <= 2.1e-177)
		tmp = Float64(-z);
	elseif (x <= 9.4e+21)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	t_2 = log(t) - y;
	tmp = 0.0;
	if (x <= -4.7e+67)
		tmp = t_1;
	elseif (x <= -1.7e-188)
		tmp = t_2;
	elseif (x <= -2.45e-256)
		tmp = -z;
	elseif (x <= 4e-255)
		tmp = t_2;
	elseif (x <= 2.1e-177)
		tmp = -z;
	elseif (x <= 9.4e+21)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[x, -4.7e+67], t$95$1, If[LessEqual[x, -1.7e-188], t$95$2, If[LessEqual[x, -2.45e-256], (-z), If[LessEqual[x, 4e-255], t$95$2, If[LessEqual[x, 2.1e-177], (-z), If[LessEqual[x, 9.4e+21], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.70000000000000017e67 or 9.4e21 < x

   1. Initial program 99.5%

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

      1. associate--l- 99.5%

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

      2. associate-+l- 99.5%

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

      3. *-commutative 99.5%

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

      4. add-cube-cbrt 98.3%

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

      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 98.3%

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

      7. pow2 98.3%

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

   3. Applied egg-rr 98.3%

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

   4. Taylor expanded in x around inf 71.6%

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

      1. pow-base-1 71.6%

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

      2. associate-*r* 71.6%

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

      3. *-lft-identity 71.6%

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

   6. Simplified 71.6%

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

   if -4.70000000000000017e67 < x < -1.70000000000000014e-188 or -2.44999999999999998e-256 < x < 4e-255 or 2.10000000000000001e-177 < x < 9.4e21

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 67.7%

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

   3. Taylor expanded in x around 0 65.5%

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

   if -1.70000000000000014e-188 < x < -2.44999999999999998e-256 or 4e-255 < x < 2.10000000000000001e-177

   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. add-cube-cbrt 100.0%

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

      4. associate-*l* 100.0%

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

      5. fma-neg 100.0%

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

      6. pow2 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in z around inf 71.6%

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

      1. neg-mul-1 71.6%

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

   6. Simplified 71.6%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+67}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-188}:\\ \;\;\;\;\log t - y\\ \mathbf{elif}\;x \leq -2.45 \cdot 10^{-256}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-255}:\\ \;\;\;\;\log t - y\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-177}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 9.4 \cdot 10^{+21}:\\ \;\;\;\;\log t - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]

Alternative 3: 47.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -3.7 \cdot 10^{+67}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-176}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{+22}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -3.7e+67)
     t_1
     (if (<= x 2.7e-176) (- z) (if (<= x 2.85e+22) (- y) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -3.7e+67) {
		tmp = t_1;
	} else if (x <= 2.7e-176) {
		tmp = -z;
	} else if (x <= 2.85e+22) {
		tmp = -y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if (x <= (-3.7d+67)) then
        tmp = t_1
    else if (x <= 2.7d-176) then
        tmp = -z
    else if (x <= 2.85d+22) then
        tmp = -y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double tmp;
	if (x <= -3.7e+67) {
		tmp = t_1;
	} else if (x <= 2.7e-176) {
		tmp = -z;
	} else if (x <= 2.85e+22) {
		tmp = -y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -3.7e+67:
		tmp = t_1
	elif x <= 2.7e-176:
		tmp = -z
	elif x <= 2.85e+22:
		tmp = -y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -3.7e+67)
		tmp = t_1;
	elseif (x <= 2.7e-176)
		tmp = Float64(-z);
	elseif (x <= 2.85e+22)
		tmp = Float64(-y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (x <= -3.7e+67)
		tmp = t_1;
	elseif (x <= 2.7e-176)
		tmp = -z;
	elseif (x <= 2.85e+22)
		tmp = -y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.7e+67], t$95$1, If[LessEqual[x, 2.7e-176], (-z), If[LessEqual[x, 2.85e+22], (-y), t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.6999999999999997e67 or 2.8499999999999999e22 < x

   1. Initial program 99.5%

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

      1. associate--l- 99.5%

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

      2. associate-+l- 99.5%

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

      3. *-commutative 99.5%

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

      4. add-cube-cbrt 98.3%

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

      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 98.3%

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

      7. pow2 98.3%

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

   3. Applied egg-rr 98.3%

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

   4. Taylor expanded in x around inf 71.6%

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

      1. pow-base-1 71.6%

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

      2. associate-*r* 71.6%

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

      3. *-lft-identity 71.6%

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

   6. Simplified 71.6%

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

   if -3.6999999999999997e67 < x < 2.6999999999999998e-176

   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. add-cube-cbrt 99.9%

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

      4. associate-*l* 100.0%

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

      5. fma-neg 100.0%

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

      6. pow2 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in z around inf 46.4%

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

      1. neg-mul-1 46.4%

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

   6. Simplified 46.4%

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

   if 2.6999999999999998e-176 < x < 2.8499999999999999e22

   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. add-cube-cbrt 100.0%

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

      4. associate-*l* 100.0%

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

      5. fma-neg 100.0%

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

      6. pow2 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in y around inf 52.0%

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

      1. neg-mul-1 52.0%

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

   6. Simplified 52.0%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+67}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-176}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{+22}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]

Alternative 4: 60.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{+64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-174}:\\ \;\;\;\;\log t - z\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+22}:\\ \;\;\;\;\log t - y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -2.3e+64)
     t_1
     (if (<= x 2.1e-174)
       (- (log t) z)
       (if (<= x 2.2e+22) (- (log t) y) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -2.3e+64) {
		tmp = t_1;
	} else if (x <= 2.1e-174) {
		tmp = log(t) - z;
	} else if (x <= 2.2e+22) {
		tmp = log(t) - y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if (x <= (-2.3d+64)) then
        tmp = t_1
    else if (x <= 2.1d-174) then
        tmp = log(t) - z
    else if (x <= 2.2d+22) then
        tmp = log(t) - y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double tmp;
	if (x <= -2.3e+64) {
		tmp = t_1;
	} else if (x <= 2.1e-174) {
		tmp = Math.log(t) - z;
	} else if (x <= 2.2e+22) {
		tmp = Math.log(t) - y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -2.3e+64:
		tmp = t_1
	elif x <= 2.1e-174:
		tmp = math.log(t) - z
	elif x <= 2.2e+22:
		tmp = math.log(t) - y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -2.3e+64)
		tmp = t_1;
	elseif (x <= 2.1e-174)
		tmp = Float64(log(t) - z);
	elseif (x <= 2.2e+22)
		tmp = Float64(log(t) - y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (x <= -2.3e+64)
		tmp = t_1;
	elseif (x <= 2.1e-174)
		tmp = log(t) - z;
	elseif (x <= 2.2e+22)
		tmp = log(t) - y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.3e+64], t$95$1, If[LessEqual[x, 2.1e-174], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, 2.2e+22], N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.3e64 or 2.2e22 < x

   1. Initial program 99.5%

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

      1. associate--l- 99.5%

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

      2. associate-+l- 99.5%

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

      3. *-commutative 99.5%

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

      4. add-cube-cbrt 98.3%

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

      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 98.3%

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

      7. pow2 98.3%

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

   3. Applied egg-rr 98.3%

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

   4. Taylor expanded in x around inf 71.6%

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

      1. pow-base-1 71.6%

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

      2. associate-*r* 71.6%

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

      3. *-lft-identity 71.6%

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

   6. Simplified 71.6%

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

   if -2.3e64 < x < 2.1000000000000001e-174

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 68.6%

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

      1. +-commutative 68.6%

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

      2. *-commutative 68.6%

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

      3. log-pow 60.3%

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

      4. log-prod 60.3%

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

   4. Simplified 60.3%

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

   5. Taylor expanded in x around 0 66.4%

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

   if 2.1000000000000001e-174 < x < 2.2e22

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 71.5%

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

   3. Taylor expanded in x around 0 71.5%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{+64}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-174}:\\ \;\;\;\;\log t - z\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+22}:\\ \;\;\;\;\log t - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]

Alternative 5: 68.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-61} \lor \neg \left(x \leq 5 \cdot 10^{-68}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\log t - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.9e-61) (not (<= x 5e-68)))
   (- (* x (log y)) y)
   (- (log t) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.9e-61) || !(x <= 5e-68)) {
		tmp = (x * log(y)) - y;
	} else {
		tmp = log(t) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-1.9d-61)) .or. (.not. (x <= 5d-68))) then
        tmp = (x * log(y)) - y
    else
        tmp = log(t) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.9e-61) || !(x <= 5e-68)) {
		tmp = (x * Math.log(y)) - y;
	} else {
		tmp = Math.log(t) - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.9e-61) or not (x <= 5e-68):
		tmp = (x * math.log(y)) - y
	else:
		tmp = math.log(t) - z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.9e-61) || !(x <= 5e-68))
		tmp = Float64(Float64(x * log(y)) - y);
	else
		tmp = Float64(log(t) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.9e-61) || ~((x <= 5e-68)))
		tmp = (x * log(y)) - y;
	else
		tmp = log(t) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.9e-61], N[Not[LessEqual[x, 5e-68]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8999999999999999e-61 or 4.99999999999999971e-68 < x

   1. Initial program 99.6%

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

   2. Taylor expanded in z around 0 79.6%

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

   3. Taylor expanded in x around inf 78.4%

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

   if -1.8999999999999999e-61 < x < 4.99999999999999971e-68

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 72.0%

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

      1. +-commutative 72.0%

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

      2. *-commutative 72.0%

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

      3. log-pow 72.0%

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

      4. log-prod 72.0%

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

   4. Simplified 72.0%

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

   5. Taylor expanded in x around 0 72.0%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-61} \lor \neg \left(x \leq 5 \cdot 10^{-68}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\log t - z\\ \end{array} \]

Alternative 6: 89.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{+67}:\\ \;\;\;\;t_1 - z\\ \mathbf{elif}\;x \leq 1960:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;t_1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -1.7e+67)
     (- t_1 z)
     (if (<= x 1960.0) (- (- (log t) y) z) (- t_1 y)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -1.7e+67) {
		tmp = t_1 - z;
	} else if (x <= 1960.0) {
		tmp = (log(t) - y) - z;
	} else {
		tmp = t_1 - y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if (x <= (-1.7d+67)) then
        tmp = t_1 - z
    else if (x <= 1960.0d0) then
        tmp = (log(t) - y) - z
    else
        tmp = t_1 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double tmp;
	if (x <= -1.7e+67) {
		tmp = t_1 - z;
	} else if (x <= 1960.0) {
		tmp = (Math.log(t) - y) - z;
	} else {
		tmp = t_1 - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -1.7e+67:
		tmp = t_1 - z
	elif x <= 1960.0:
		tmp = (math.log(t) - y) - z
	else:
		tmp = t_1 - y
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -1.7e+67)
		tmp = Float64(t_1 - z);
	elseif (x <= 1960.0)
		tmp = Float64(Float64(log(t) - y) - z);
	else
		tmp = Float64(t_1 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (x <= -1.7e+67)
		tmp = t_1 - z;
	elseif (x <= 1960.0)
		tmp = (log(t) - y) - z;
	else
		tmp = t_1 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.7e+67], N[(t$95$1 - z), $MachinePrecision], If[LessEqual[x, 1960.0], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], N[(t$95$1 - y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.7000000000000001e67

   1. Initial program 99.5%

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

   2. Taylor expanded in y around 0 89.6%

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

   3. Taylor expanded in x around inf 89.6%

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

   if -1.7000000000000001e67 < x < 1960

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.3%

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

      1. associate--r+ 98.3%

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

   4. Simplified 98.3%

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

   if 1960 < x

   1. Initial program 99.5%

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

   2. Taylor expanded in z around 0 88.3%

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

   3. Taylor expanded in x around inf 88.3%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+67}:\\ \;\;\;\;x \cdot \log y - z\\ \mathbf{elif}\;x \leq 1960:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - y\\ \end{array} \]

Alternative 7: 75.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;y \leq 1.04 \cdot 10^{+127}:\\ \;\;\;\;t_1 - z\\ \mathbf{else}:\\ \;\;\;\;t_1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y)))) (if (<= y 1.04e+127) (- t_1 z) (- t_1 y))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (y <= 1.04e+127) {
		tmp = t_1 - z;
	} else {
		tmp = t_1 - y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if (y <= 1.04d+127) then
        tmp = t_1 - z
    else
        tmp = t_1 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double tmp;
	if (y <= 1.04e+127) {
		tmp = t_1 - z;
	} else {
		tmp = t_1 - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if y <= 1.04e+127:
		tmp = t_1 - z
	else:
		tmp = t_1 - y
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (y <= 1.04e+127)
		tmp = Float64(t_1 - z);
	else
		tmp = Float64(t_1 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (y <= 1.04e+127)
		tmp = t_1 - z;
	else
		tmp = t_1 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.04e+127], N[(t$95$1 - z), $MachinePrecision], N[(t$95$1 - y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.04e127

   1. Initial program 99.7%

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

   2. Taylor expanded in y around 0 91.4%

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

   3. Taylor expanded in x around inf 75.7%

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

   if 1.04e127 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 89.3%

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

   3. Taylor expanded in x around inf 89.3%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.04 \cdot 10^{+127}:\\ \;\;\;\;x \cdot \log y - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - y\\ \end{array} \]

Alternative 8: 46.4% accurate, 51.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{+133}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 1.15e+133) (- z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.15e+133) {
		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 <= 1.15d+133) 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 <= 1.15e+133) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 1.15e+133:
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.15e+133)
		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 <= 1.15e+133)
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.15e+133], (-z), (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 1.14999999999999995e133

   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. add-cube-cbrt 99.2%

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

      4. associate-*l* 99.2%

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

      5. fma-neg 99.2%

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

      6. pow2 99.2%

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

   3. Applied egg-rr 99.2%

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

   4. Taylor expanded in z around inf 37.3%

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

      1. neg-mul-1 37.3%

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

   6. Simplified 37.3%

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

   if 1.14999999999999995e133 < y

   1. Initial program 99.9%

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

      1. associate--l- 99.9%

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

      2. associate-+l- 99.9%

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

      3. add-cube-cbrt 99.6%

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

      4. associate-*l* 99.6%

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

      5. fma-neg 99.6%

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

      6. pow2 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in y around inf 75.4%

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

      1. neg-mul-1 75.4%

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

   6. Simplified 75.4%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{+133}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 9: 30.5% accurate, 104.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate--l- 99.8%

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

   2. associate-+l- 99.8%

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

   3. add-cube-cbrt 99.3%

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

   4. associate-*l* 99.3%

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

   5. fma-neg 99.3%

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

   6. pow2 99.3%

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

3. Applied egg-rr 99.3%

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

4. Taylor expanded in y around inf 26.7%

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

   1. neg-mul-1 26.7%

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

6. Simplified 26.7%

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

7. Final simplification 26.7%

   \[\leadsto -y \]

Reproduce

herbie shell --seed 2023199 
(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)))