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

Percentage Accurate: 99.9% → 99.9%

Time: 10.7s

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 99.9%

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

Alternative 2: 88.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.65 \cdot 10^{+47} \lor \neg \left(z \leq 1.25 \cdot 10^{+149}\right):\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log y + \log t\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.65e+47) (not (<= z 1.25e+149)))
   (- (log t) (+ y z))
   (- (+ (* x (log y)) (log t)) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.65e+47) || !(z <= 1.25e+149)) {
		tmp = log(t) - (y + z);
	} else {
		tmp = ((x * log(y)) + 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 <= (-2.65d+47)) .or. (.not. (z <= 1.25d+149))) then
        tmp = log(t) - (y + z)
    else
        tmp = ((x * log(y)) + 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 <= -2.65e+47) || !(z <= 1.25e+149)) {
		tmp = Math.log(t) - (y + z);
	} else {
		tmp = ((x * Math.log(y)) + Math.log(t)) - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2.65e+47) or not (z <= 1.25e+149):
		tmp = math.log(t) - (y + z)
	else:
		tmp = ((x * math.log(y)) + math.log(t)) - y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.65e+47) || !(z <= 1.25e+149))
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = Float64(Float64(Float64(x * log(y)) + log(t)) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -2.65e+47) || ~((z <= 1.25e+149)))
		tmp = log(t) - (y + z);
	else
		tmp = ((x * log(y)) + log(t)) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2.65e+47], N[Not[LessEqual[z, 1.25e+149]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[Log[t], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.65e47 or 1.24999999999999998e149 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 89.1%

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

   if -2.65e47 < z < 1.24999999999999998e149

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 96.0%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.65 \cdot 10^{+47} \lor \neg \left(z \leq 1.25 \cdot 10^{+149}\right):\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log y + \log t\right) - y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+200} \lor \neg \left(x \leq 1.85 \cdot 10^{+157}\right):\\ \;\;\;\;x \cdot \log y + \log t\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -5.2e+200) (not (<= x 1.85e+157)))
   (+ (* x (log y)) (log t))
   (- (log t) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -5.2e+200) || !(x <= 1.85e+157)) {
		tmp = (x * log(y)) + log(t);
	} 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 <= (-5.2d+200)) .or. (.not. (x <= 1.85d+157))) then
        tmp = (x * log(y)) + log(t)
    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 <= -5.2e+200) || !(x <= 1.85e+157)) {
		tmp = (x * Math.log(y)) + Math.log(t);
	} else {
		tmp = Math.log(t) - (y + z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -5.2e+200) or not (x <= 1.85e+157):
		tmp = (x * math.log(y)) + math.log(t)
	else:
		tmp = math.log(t) - (y + z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -5.2e+200) || !(x <= 1.85e+157))
		tmp = Float64(Float64(x * log(y)) + log(t));
	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 <= -5.2e+200) || ~((x <= 1.85e+157)))
		tmp = (x * log(y)) + log(t);
	else
		tmp = log(t) - (y + z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -5.2e+200], N[Not[LessEqual[x, 1.85e+157]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[Log[t], $MachinePrecision]), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.2000000000000003e200 or 1.8499999999999999e157 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf 59.3%

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

      1. associate--r+ 59.3%

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

      2. mul-1-neg 59.3%

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

      3. log-rec 59.3%

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

      4. mul-1-neg 59.3%

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

      5. associate-/l* 59.3%

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

      6. distribute-rgt-neg-in 59.3%

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

      7. fma-neg 59.3%

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

      8. mul-1-neg 59.3%

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

      9. distribute-frac-neg 59.3%

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

      10. remove-double-neg 59.3%

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

      11. metadata-eval 59.3%

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

   5. Simplified 59.3%

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

   6. Taylor expanded in z around -inf 45.0%

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

   7. Taylor expanded in x around inf 80.1%

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

   if -5.2000000000000003e200 < x < 1.8499999999999999e157

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 88.3%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+200} \lor \neg \left(x \leq 1.85 \cdot 10^{+157}\right):\\ \;\;\;\;x \cdot \log y + \log t\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{+27}:\\ \;\;\;\;\left(x \cdot \log y + \log t\right) - z\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.9e+27) (- (+ (* x (log y)) (log t)) z) (- (log t) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.9e+27) {
		tmp = ((x * log(y)) + log(t)) - 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 (y <= 1.9d+27) then
        tmp = ((x * log(y)) + log(t)) - 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 (y <= 1.9e+27) {
		tmp = ((x * Math.log(y)) + Math.log(t)) - z;
	} else {
		tmp = Math.log(t) - (y + z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 1.9e+27:
		tmp = ((x * math.log(y)) + math.log(t)) - z
	else:
		tmp = math.log(t) - (y + z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.9e+27)
		tmp = Float64(Float64(Float64(x * log(y)) + log(t)) - 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 (y <= 1.9e+27)
		tmp = ((x * log(y)) + log(t)) - z;
	else
		tmp = log(t) - (y + z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.9e+27], N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[Log[t], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.90000000000000011e27

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 99.2%

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

   if 1.90000000000000011e27 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 84.5%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{+27}:\\ \;\;\;\;\left(x \cdot \log y + \log t\right) - z\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 47.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-298}:\\ \;\;\;\;\log t\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-95}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-46}:\\ \;\;\;\;\log t\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+44}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 3.3e-298)
   (log t)
   (if (<= y 2.5e-95)
     (- z)
     (if (<= y 1.8e-46) (log t) (if (<= y 5.2e+44) (- z) (- y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.3e-298) {
		tmp = log(t);
	} else if (y <= 2.5e-95) {
		tmp = -z;
	} else if (y <= 1.8e-46) {
		tmp = log(t);
	} else if (y <= 5.2e+44) {
		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 <= 3.3d-298) then
        tmp = log(t)
    else if (y <= 2.5d-95) then
        tmp = -z
    else if (y <= 1.8d-46) then
        tmp = log(t)
    else if (y <= 5.2d+44) 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 <= 3.3e-298) {
		tmp = Math.log(t);
	} else if (y <= 2.5e-95) {
		tmp = -z;
	} else if (y <= 1.8e-46) {
		tmp = Math.log(t);
	} else if (y <= 5.2e+44) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 3.3e-298:
		tmp = math.log(t)
	elif y <= 2.5e-95:
		tmp = -z
	elif y <= 1.8e-46:
		tmp = math.log(t)
	elif y <= 5.2e+44:
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3.3e-298)
		tmp = log(t);
	elseif (y <= 2.5e-95)
		tmp = Float64(-z);
	elseif (y <= 1.8e-46)
		tmp = log(t);
	elseif (y <= 5.2e+44)
		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 <= 3.3e-298)
		tmp = log(t);
	elseif (y <= 2.5e-95)
		tmp = -z;
	elseif (y <= 1.8e-46)
		tmp = log(t);
	elseif (y <= 5.2e+44)
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 3.3e-298], N[Log[t], $MachinePrecision], If[LessEqual[y, 2.5e-95], (-z), If[LessEqual[y, 1.8e-46], N[Log[t], $MachinePrecision], If[LessEqual[y, 5.2e+44], (-z), (-y)]]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.3000000000000002e-298 or 2.4999999999999999e-95 < y < 1.8e-46

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 90.8%

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

      1. associate--r+ 90.8%

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

      2. mul-1-neg 90.8%

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

      3. log-rec 90.8%

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

      4. mul-1-neg 90.8%

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

      5. associate-/l* 90.8%

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

      6. distribute-rgt-neg-in 90.8%

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

      7. fma-neg 90.8%

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

      8. mul-1-neg 90.8%

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

      9. distribute-frac-neg 90.8%

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

      10. remove-double-neg 90.8%

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

      11. metadata-eval 90.8%

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

   5. Simplified 90.8%

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

   6. Taylor expanded in x around inf 91.2%

      \[\leadsto y \cdot \color{blue}{\frac{x \cdot \log y}{y}} + \log t \]
   7. Step-by-step derivation

      1. associate-*r/ 91.2%

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

   8. Simplified 91.2%

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

   9. Taylor expanded in x around 0 62.7%

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

   if 3.3000000000000002e-298 < y < 2.4999999999999999e-95 or 1.8e-46 < y < 5.1999999999999998e44

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf 43.8%

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

      1. neg-mul-1 43.8%

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

   5. Simplified 43.8%

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

   if 5.1999999999999998e44 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 70.4%

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

      1. neg-mul-1 70.4%

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

   5. Simplified 70.4%

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

Alternative 6: 60.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+44}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 6e+44) (- (log t) z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 6e+44) {
		tmp = log(t) - z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 6d+44) then
        tmp = log(t) - z
    else
        tmp = -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 6e+44) {
		tmp = Math.log(t) - z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 6e+44:
		tmp = math.log(t) - z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 6e+44)
		tmp = Float64(log(t) - z);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 6e+44)
		tmp = log(t) - z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 6e+44], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 5.99999999999999974e44

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 97.8%

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

   4. Taylor expanded in x around 0 64.3%

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

   if 5.99999999999999974e44 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 70.4%

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

      1. neg-mul-1 70.4%

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

   5. Simplified 70.4%

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

Alternative 7: 70.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0 74.8%

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

Alternative 8: 48.6% accurate, 29.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+44}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 5e+44) (- z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 5e+44) {
		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 <= 5d+44) 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 <= 5e+44) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 5e+44:
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 5e+44)
		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 <= 5e+44)
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 5e+44], (-z), (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 4.9999999999999996e44

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf 38.1%

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

      1. neg-mul-1 38.1%

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

   5. Simplified 38.1%

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

   if 4.9999999999999996e44 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 70.4%

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

      1. neg-mul-1 70.4%

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

   5. Simplified 70.4%

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

Alternative 9: 30.1% 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. Add Preprocessing

3. Taylor expanded in y around inf 34.0%

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

   1. neg-mul-1 34.0%

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

5. Simplified 34.0%

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

Reproduce

herbie shell --seed 2024096 
(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)))