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

Percentage Accurate: 99.9% → 99.8%

Time: 12.0s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\log t - z\right) - y\right) - x \cdot \log \left(\frac{1}{y}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (- (- (log t) z) y) (* x (log (/ 1.0 y)))))

C

double code(double x, double y, double z, double t) {
	return ((log(t) - z) - y) - (x * log((1.0 / y)));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((log(t) - z) - y) - (x * log((1.0d0 / y)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((Math.log(t) - z) - y) - (x * Math.log((1.0 / y)));
}

Python

def code(x, y, z, t):
	return ((math.log(t) - z) - y) - (x * math.log((1.0 / y)))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(log(t) - z) - y) - Float64(x * log(Float64(1.0 / y))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((log(t) - z) - y) - (x * log((1.0 / y)));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision] - y), $MachinePrecision] - N[(x * N[Log[N[(1.0 / y), $MachinePrecision]], $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. associate-+l- 99.9%

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

   2. associate--l- 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in y around inf 99.9%

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

6. Final simplification 99.9%

   \[\leadsto \left(\left(\log t - z\right) - y\right) - x \cdot \log \left(\frac{1}{y}\right) \]
7. Add Preprocessing

Alternative 2: 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. Add Preprocessing

3. Final simplification 99.9%

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

Alternative 3: 61.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-131}:\\ \;\;\;\;\log t - y\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+76}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -3.8e+92)
     t_1
     (if (<= x -1.5e-131)
       (- (log t) y)
       (if (<= x 1.35e+76) (- (log t) z) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -3.8e+92) {
		tmp = t_1;
	} else if (x <= -1.5e-131) {
		tmp = log(t) - y;
	} else if (x <= 1.35e+76) {
		tmp = log(t) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if (x <= (-3.8d+92)) then
        tmp = t_1
    else if (x <= (-1.5d-131)) then
        tmp = log(t) - y
    else if (x <= 1.35d+76) then
        tmp = log(t) - z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double tmp;
	if (x <= -3.8e+92) {
		tmp = t_1;
	} else if (x <= -1.5e-131) {
		tmp = Math.log(t) - y;
	} else if (x <= 1.35e+76) {
		tmp = Math.log(t) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -3.8e+92:
		tmp = t_1
	elif x <= -1.5e-131:
		tmp = math.log(t) - y
	elif x <= 1.35e+76:
		tmp = math.log(t) - z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -3.8e+92)
		tmp = t_1;
	elseif (x <= -1.5e-131)
		tmp = Float64(log(t) - y);
	elseif (x <= 1.35e+76)
		tmp = Float64(log(t) - z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (x <= -3.8e+92)
		tmp = t_1;
	elseif (x <= -1.5e-131)
		tmp = log(t) - y;
	elseif (x <= 1.35e+76)
		tmp = log(t) - z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.8e+92], t$95$1, If[LessEqual[x, -1.5e-131], N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision], If[LessEqual[x, 1.35e+76], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.8e92 or 1.34999999999999995e76 < x

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. associate--l+ 99.6%

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

      3. fma-define 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in z around inf 65.2%

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

      1. associate--r+ 65.2%

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

      2. associate-/l* 65.1%

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

   7. Simplified 65.1%

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

      1. clear-num 65.0%

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

      2. un-div-inv 65.1%

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

   9. Applied egg-rr 65.1%

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

       1. clear-num 65.1%

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

       2. inv-pow 65.1%

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

   11. Applied egg-rr 65.1%

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

       1. unpow-1 65.1%

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

   13. Simplified 65.1%

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

   14. Taylor expanded in x around inf 70.3%

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

   if -3.8e92 < x < -1.49999999999999998e-131

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. associate--l+ 100.0%

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

      3. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 72.6%

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

      1. mul-1-neg 72.6%

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

   7. Simplified 72.6%

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

   8. Taylor expanded in y around 0 72.6%

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

      1. neg-mul-1 72.6%

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

      2. sub-neg 72.6%

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

   10. Simplified 72.6%

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

   if -1.49999999999999998e-131 < x < 1.34999999999999995e76

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. associate--l+ 100.0%

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

      3. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 63.3%

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

      1. neg-mul-1 63.3%

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

   7. Simplified 63.3%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+92}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-131}:\\ \;\;\;\;\log t - y\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+76}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-129}:\\ \;\;\;\;-y\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+76}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -1.8e+91)
     t_1
     (if (<= x -1.45e-129) (- y) (if (<= x 1.35e+76) (- z) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -1.8e+91) {
		tmp = t_1;
	} else if (x <= -1.45e-129) {
		tmp = -y;
	} else if (x <= 1.35e+76) {
		tmp = -z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if (x <= (-1.8d+91)) then
        tmp = t_1
    else if (x <= (-1.45d-129)) then
        tmp = -y
    else if (x <= 1.35d+76) then
        tmp = -z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double tmp;
	if (x <= -1.8e+91) {
		tmp = t_1;
	} else if (x <= -1.45e-129) {
		tmp = -y;
	} else if (x <= 1.35e+76) {
		tmp = -z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -1.8e+91:
		tmp = t_1
	elif x <= -1.45e-129:
		tmp = -y
	elif x <= 1.35e+76:
		tmp = -z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -1.8e+91)
		tmp = t_1;
	elseif (x <= -1.45e-129)
		tmp = Float64(-y);
	elseif (x <= 1.35e+76)
		tmp = Float64(-z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (x <= -1.8e+91)
		tmp = t_1;
	elseif (x <= -1.45e-129)
		tmp = -y;
	elseif (x <= 1.35e+76)
		tmp = -z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.8e+91], t$95$1, If[LessEqual[x, -1.45e-129], (-y), If[LessEqual[x, 1.35e+76], (-z), t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.8e91 or 1.34999999999999995e76 < x

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. associate--l+ 99.6%

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

      3. fma-define 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in z around inf 65.2%

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

      1. associate--r+ 65.2%

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

      2. associate-/l* 65.1%

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

   7. Simplified 65.1%

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

      1. clear-num 65.0%

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

      2. un-div-inv 65.1%

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

   9. Applied egg-rr 65.1%

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

       1. clear-num 65.1%

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

       2. inv-pow 65.1%

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

   11. Applied egg-rr 65.1%

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

       1. unpow-1 65.1%

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

   13. Simplified 65.1%

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

   14. Taylor expanded in x around inf 70.3%

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

   if -1.8e91 < x < -1.45000000000000008e-129

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. associate--l+ 100.0%

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

      3. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 72.6%

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

      1. mul-1-neg 72.6%

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

   7. Simplified 72.6%

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

   8. Taylor expanded in y around inf 57.1%

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

      1. neg-mul-1 57.1%

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

   10. Simplified 57.1%

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

   if -1.45000000000000008e-129 < x < 1.34999999999999995e76

   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%

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

      2. associate--l- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 48.9%

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

   6. Taylor expanded in x around 0 44.9%

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

      1. neg-mul-1 44.9%

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

   8. Simplified 44.9%

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

Alternative 5: 89.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+91} \lor \neg \left(x \leq 3.3 \cdot 10^{+63}\right):\\ \;\;\;\;x \cdot \log y - z\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -3.1e+91) (not (<= x 3.3e+63)))
   (- (* x (log y)) z)
   (- (log t) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.1e+91) || !(x <= 3.3e+63)) {
		tmp = (x * log(y)) - z;
	} else {
		tmp = log(t) - (y + z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-3.1d+91)) .or. (.not. (x <= 3.3d+63))) then
        tmp = (x * log(y)) - z
    else
        tmp = log(t) - (y + z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.1e+91) || !(x <= 3.3e+63)) {
		tmp = (x * Math.log(y)) - z;
	} else {
		tmp = Math.log(t) - (y + z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -3.1e+91) or not (x <= 3.3e+63):
		tmp = (x * math.log(y)) - z
	else:
		tmp = math.log(t) - (y + z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -3.1e+91) || !(x <= 3.3e+63))
		tmp = Float64(Float64(x * log(y)) - z);
	else
		tmp = Float64(log(t) - Float64(y + z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -3.1e+91) || ~((x <= 3.3e+63)))
		tmp = (x * log(y)) - z;
	else
		tmp = log(t) - (y + z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -3.1e+91], N[Not[LessEqual[x, 3.3e+63]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.09999999999999998e91 or 3.3000000000000002e63 < x

   1. Initial program 99.6%

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

      1. associate-+l- 99.6%

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

      2. associate--l- 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in z around inf 86.9%

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

   if -3.09999999999999998e91 < x < 3.3000000000000002e63

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. associate--l+ 100.0%

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

      3. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 95.5%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+91} \lor \neg \left(x \leq 3.3 \cdot 10^{+63}\right):\\ \;\;\;\;x \cdot \log y - z\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+93} \lor \neg \left(x \leq 3.6 \cdot 10^{+75}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.8e+93) (not (<= x 3.6e+75)))
   (- (* x (log y)) y)
   (- (log t) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.8e+93) || !(x <= 3.6e+75)) {
		tmp = (x * log(y)) - 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 <= (-2.8d+93)) .or. (.not. (x <= 3.6d+75))) then
        tmp = (x * log(y)) - 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 <= -2.8e+93) || !(x <= 3.6e+75)) {
		tmp = (x * Math.log(y)) - y;
	} else {
		tmp = Math.log(t) - (y + z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2.8e+93) or not (x <= 3.6e+75):
		tmp = (x * math.log(y)) - y
	else:
		tmp = math.log(t) - (y + z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.8e+93) || !(x <= 3.6e+75))
		tmp = Float64(Float64(x * log(y)) - y);
	else
		tmp = Float64(log(t) - Float64(y + z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -2.8e+93) || ~((x <= 3.6e+75)))
		tmp = (x * log(y)) - y;
	else
		tmp = log(t) - (y + z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.8e+93], N[Not[LessEqual[x, 3.6e+75]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.79999999999999989e93 or 3.6e75 < x

   1. Initial program 99.6%

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

      1. associate-+l- 99.6%

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

      2. associate--l- 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 83.6%

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

   if -2.79999999999999989e93 < x < 3.6e75

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. associate--l+ 100.0%

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

      3. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 95.1%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+93} \lor \neg \left(x \leq 3.6 \cdot 10^{+75}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+91} \lor \neg \left(x \leq 5.9 \cdot 10^{+190}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.8e+91) (not (<= x 5.9e+190)))
   (* x (log y))
   (- (log t) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.8e+91) || !(x <= 5.9e+190)) {
		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 <= (-2.8d+91)) .or. (.not. (x <= 5.9d+190))) 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 <= -2.8e+91) || !(x <= 5.9e+190)) {
		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 <= -2.8e+91) or not (x <= 5.9e+190):
		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 <= -2.8e+91) || !(x <= 5.9e+190))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(log(t) - Float64(y + z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -2.8e+91) || ~((x <= 5.9e+190)))
		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, -2.8e+91], N[Not[LessEqual[x, 5.9e+190]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.7999999999999999e91 or 5.89999999999999972e190 < x

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. associate--l+ 99.6%

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

      3. fma-define 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in z around inf 63.6%

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

      1. associate--r+ 63.6%

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

      2. associate-/l* 63.5%

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

   7. Simplified 63.5%

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

      1. clear-num 63.3%

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

      2. un-div-inv 63.5%

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

   9. Applied egg-rr 63.5%

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

       1. clear-num 63.5%

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

       2. inv-pow 63.5%

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

   11. Applied egg-rr 63.5%

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

       1. unpow-1 63.5%

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

   13. Simplified 63.5%

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

   14. Taylor expanded in x around inf 76.4%

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

   if -2.7999999999999999e91 < x < 5.89999999999999972e190

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. associate--l+ 100.0%

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

      3. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 90.5%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+91} \lor \neg \left(x \leq 5.9 \cdot 10^{+190}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+108}:\\ \;\;\;\;\log \left(\frac{1}{y}\right) \cdot \left(-x\right) - z\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+63}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.6e+108)
   (- (* (log (/ 1.0 y)) (- x)) z)
   (if (<= x 3.3e+63) (- (log t) (+ y z)) (- (* x (log y)) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.6e+108) {
		tmp = (log((1.0 / y)) * -x) - z;
	} else if (x <= 3.3e+63) {
		tmp = log(t) - (y + z);
	} else {
		tmp = (x * log(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 <= (-1.6d+108)) then
        tmp = (log((1.0d0 / y)) * -x) - z
    else if (x <= 3.3d+63) then
        tmp = log(t) - (y + z)
    else
        tmp = (x * log(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 <= -1.6e+108) {
		tmp = (Math.log((1.0 / y)) * -x) - z;
	} else if (x <= 3.3e+63) {
		tmp = Math.log(t) - (y + z);
	} else {
		tmp = (x * Math.log(y)) - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.6e+108:
		tmp = (math.log((1.0 / y)) * -x) - z
	elif x <= 3.3e+63:
		tmp = math.log(t) - (y + z)
	else:
		tmp = (x * math.log(y)) - z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.6e+108)
		tmp = Float64(Float64(log(Float64(1.0 / y)) * Float64(-x)) - z);
	elseif (x <= 3.3e+63)
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = Float64(Float64(x * log(y)) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.6e+108)
		tmp = (log((1.0 / y)) * -x) - z;
	elseif (x <= 3.3e+63)
		tmp = log(t) - (y + z);
	else
		tmp = (x * log(y)) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.6e+108], N[(N[(N[Log[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] * (-x)), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, 3.3e+63], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6e108

   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%

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

      2. associate--l- 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in z around inf 86.9%

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

   6. Taylor expanded in y around inf 87.0%

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

   if -1.6e108 < x < 3.3000000000000002e63

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. associate--l+ 100.0%

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

      3. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 95.5%

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

   if 3.3000000000000002e63 < x

   1. Initial program 99.8%

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

      1. associate-+l- 99.8%

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

      2. associate--l- 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 86.8%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+108}:\\ \;\;\;\;\log \left(\frac{1}{y}\right) \cdot \left(-x\right) - z\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+63}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+22} \lor \neg \left(z \leq 20000000\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\log t - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.02e+22) (not (<= z 20000000.0))) (- z) (- (log t) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.02e+22) || !(z <= 20000000.0)) {
		tmp = -z;
	} else {
		tmp = log(t) - y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-1.02d+22)) .or. (.not. (z <= 20000000.0d0))) then
        tmp = -z
    else
        tmp = log(t) - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.02e+22) || !(z <= 20000000.0)) {
		tmp = -z;
	} else {
		tmp = Math.log(t) - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.02e+22) or not (z <= 20000000.0):
		tmp = -z
	else:
		tmp = math.log(t) - y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.02e+22) || !(z <= 20000000.0))
		tmp = Float64(-z);
	else
		tmp = Float64(log(t) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.02e+22) || ~((z <= 20000000.0)))
		tmp = -z;
	else
		tmp = log(t) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.02e+22], N[Not[LessEqual[z, 20000000.0]], $MachinePrecision]], (-z), N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.02e22 or 2e7 < z

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. associate--l- 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 81.7%

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

   6. Taylor expanded in x around 0 63.4%

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

      1. neg-mul-1 63.4%

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

   8. Simplified 63.4%

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

   if -1.02e22 < z < 2e7

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. associate--l+ 99.8%

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

      3. fma-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 65.0%

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

      1. mul-1-neg 65.0%

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

   7. Simplified 65.0%

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

   8. Taylor expanded in y around 0 65.0%

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

      1. neg-mul-1 65.0%

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

      2. sub-neg 65.0%

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

   10. Simplified 65.0%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+22} \lor \neg \left(z \leq 20000000\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\log t - y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.6% accurate, 29.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 126000000:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 126000000.0) (- z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 126000000.0) {
		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 <= 126000000.0d0) 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 <= 126000000.0) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 126000000.0:
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 126000000.0)
		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 <= 126000000.0)
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 126000000.0], (-z), (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 1.26e8

   1. Initial program 99.8%

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

      1. associate-+l- 99.8%

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

      2. associate--l- 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 76.0%

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

   6. Taylor expanded in x around 0 41.2%

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

      1. neg-mul-1 41.2%

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

   8. Simplified 41.2%

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

   if 1.26e8 < y

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. associate--l+ 99.9%

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

      3. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around inf 62.4%

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

      1. mul-1-neg 62.4%

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

   7. Simplified 62.4%

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

   8. Taylor expanded in y around inf 60.1%

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

      1. neg-mul-1 60.1%

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

   10. Simplified 60.1%

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

Alternative 11: 29.7% 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. sub-neg 99.9%

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

   2. associate--l+ 99.9%

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

   3. fma-define 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in y around inf 43.2%

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

   1. mul-1-neg 43.2%

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

7. Simplified 43.2%

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

8. Taylor expanded in y around inf 30.1%

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

   1. neg-mul-1 30.1%

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

10. Simplified 30.1%

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

Alternative 12: 2.3% accurate, 209.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. sub-neg 99.9%

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

   2. associate--l+ 99.9%

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

   3. fma-define 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in y around inf 43.2%

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

   1. mul-1-neg 43.2%

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

7. Simplified 43.2%

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

   1. *-un-lft-identity 43.2%

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

   2. add-sqr-sqrt 0.0%

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

   3. sqrt-unprod 13.6%

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

   4. sqr-neg 13.6%

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

   5. sqrt-unprod 13.5%

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

   6. add-sqr-sqrt 13.5%

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

9. Applied egg-rr 13.5%

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

    1. *-lft-identity 13.5%

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

11. Simplified 13.5%

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

12. Taylor expanded in y around inf 2.3%

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

Reproduce

herbie shell --seed 2024137 
(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)))