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

Percentage Accurate: 99.9% → 99.9%

Time: 10.3s

Alternatives: 8

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. Final simplification 99.9%

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

Alternative 2: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \lor \neg \left(z \leq 2.2 \cdot 10^{+79}\right):\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log t + x \cdot \log y\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.55) (not (<= z 2.2e+79)))
   (- (log t) (+ y z))
   (- (+ (log t) (* x (log y))) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.55) || !(z <= 2.2e+79)) {
		tmp = log(t) - (y + z);
	} else {
		tmp = (log(t) + (x * log(y))) - y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-1.55d0)) .or. (.not. (z <= 2.2d+79))) then
        tmp = log(t) - (y + z)
    else
        tmp = (log(t) + (x * log(y))) - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.55) || !(z <= 2.2e+79)) {
		tmp = Math.log(t) - (y + z);
	} else {
		tmp = (Math.log(t) + (x * Math.log(y))) - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.55) or not (z <= 2.2e+79):
		tmp = math.log(t) - (y + z)
	else:
		tmp = (math.log(t) + (x * math.log(y))) - y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.55) || !(z <= 2.2e+79))
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = Float64(Float64(log(t) + Float64(x * log(y))) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.55) || ~((z <= 2.2e+79)))
		tmp = log(t) - (y + z);
	else
		tmp = (log(t) + (x * log(y))) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.55], N[Not[LessEqual[z, 2.2e+79]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.55000000000000004 or 2.1999999999999999e79 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 92.6%

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

   if -1.55000000000000004 < z < 2.1999999999999999e79

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0 98.4%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \lor \neg \left(z \leq 2.2 \cdot 10^{+79}\right):\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log t + x \cdot \log y\right) - y\\ \end{array} \]

Alternative 3: 48.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;z \leq -3.3 \cdot 10^{+112}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{-97}:\\ \;\;\;\;-y\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-193}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{-244}:\\ \;\;\;\;-y\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-105}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{+78}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= z -3.3e+112)
     (- z)
     (if (<= z -1.6e-97)
       (- y)
       (if (<= z -1.1e-193)
         t_1
         (if (<= z -1.9e-244)
           (- y)
           (if (<= z 1.05e-105) t_1 (if (<= z 4.3e+78) (- y) (- z)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (z <= -3.3e+112) {
		tmp = -z;
	} else if (z <= -1.6e-97) {
		tmp = -y;
	} else if (z <= -1.1e-193) {
		tmp = t_1;
	} else if (z <= -1.9e-244) {
		tmp = -y;
	} else if (z <= 1.05e-105) {
		tmp = t_1;
	} else if (z <= 4.3e+78) {
		tmp = -y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if (z <= (-3.3d+112)) then
        tmp = -z
    else if (z <= (-1.6d-97)) then
        tmp = -y
    else if (z <= (-1.1d-193)) then
        tmp = t_1
    else if (z <= (-1.9d-244)) then
        tmp = -y
    else if (z <= 1.05d-105) then
        tmp = t_1
    else if (z <= 4.3d+78) then
        tmp = -y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double tmp;
	if (z <= -3.3e+112) {
		tmp = -z;
	} else if (z <= -1.6e-97) {
		tmp = -y;
	} else if (z <= -1.1e-193) {
		tmp = t_1;
	} else if (z <= -1.9e-244) {
		tmp = -y;
	} else if (z <= 1.05e-105) {
		tmp = t_1;
	} else if (z <= 4.3e+78) {
		tmp = -y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if z <= -3.3e+112:
		tmp = -z
	elif z <= -1.6e-97:
		tmp = -y
	elif z <= -1.1e-193:
		tmp = t_1
	elif z <= -1.9e-244:
		tmp = -y
	elif z <= 1.05e-105:
		tmp = t_1
	elif z <= 4.3e+78:
		tmp = -y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (z <= -3.3e+112)
		tmp = Float64(-z);
	elseif (z <= -1.6e-97)
		tmp = Float64(-y);
	elseif (z <= -1.1e-193)
		tmp = t_1;
	elseif (z <= -1.9e-244)
		tmp = Float64(-y);
	elseif (z <= 1.05e-105)
		tmp = t_1;
	elseif (z <= 4.3e+78)
		tmp = Float64(-y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (z <= -3.3e+112)
		tmp = -z;
	elseif (z <= -1.6e-97)
		tmp = -y;
	elseif (z <= -1.1e-193)
		tmp = t_1;
	elseif (z <= -1.9e-244)
		tmp = -y;
	elseif (z <= 1.05e-105)
		tmp = t_1;
	elseif (z <= 4.3e+78)
		tmp = -y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.3e+112], (-z), If[LessEqual[z, -1.6e-97], (-y), If[LessEqual[z, -1.1e-193], t$95$1, If[LessEqual[z, -1.9e-244], (-y), If[LessEqual[z, 1.05e-105], t$95$1, If[LessEqual[z, 4.3e+78], (-y), (-z)]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.2999999999999999e112 or 4.29999999999999981e78 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 86.3%

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

      1. mul-1-neg 86.3%

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

   4. Simplified 86.3%

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

   if -3.2999999999999999e112 < z < -1.5999999999999999e-97 or -1.09999999999999988e-193 < z < -1.9e-244 or 1.05e-105 < z < 4.29999999999999981e78

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 53.3%

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

      1. neg-mul-1 53.3%

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

   4. Simplified 53.3%

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

   if -1.5999999999999999e-97 < z < -1.09999999999999988e-193 or -1.9e-244 < z < 1.05e-105

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0 99.8%

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

   3. Taylor expanded in x around inf 55.7%

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

      1. *-commutative 55.7%

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

   5. Simplified 55.7%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+112}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{-97}:\\ \;\;\;\;-y\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-193}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{-244}:\\ \;\;\;\;-y\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-105}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{+78}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 4: 59.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t - y\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{+111}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{-246}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.3 \cdot 10^{-302}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+78}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (log t) y)))
   (if (<= z -3.2e+111)
     (- z)
     (if (<= z -2.3e-246)
       t_1
       (if (<= z -4.3e-302) (* x (log y)) (if (<= z 3.3e+78) t_1 (- z)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(t) - y;
	double tmp;
	if (z <= -3.2e+111) {
		tmp = -z;
	} else if (z <= -2.3e-246) {
		tmp = t_1;
	} else if (z <= -4.3e-302) {
		tmp = x * log(y);
	} else if (z <= 3.3e+78) {
		tmp = t_1;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(t) - y
    if (z <= (-3.2d+111)) then
        tmp = -z
    else if (z <= (-2.3d-246)) then
        tmp = t_1
    else if (z <= (-4.3d-302)) then
        tmp = x * log(y)
    else if (z <= 3.3d+78) then
        tmp = t_1
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(t) - y;
	double tmp;
	if (z <= -3.2e+111) {
		tmp = -z;
	} else if (z <= -2.3e-246) {
		tmp = t_1;
	} else if (z <= -4.3e-302) {
		tmp = x * Math.log(y);
	} else if (z <= 3.3e+78) {
		tmp = t_1;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(t) - y
	tmp = 0
	if z <= -3.2e+111:
		tmp = -z
	elif z <= -2.3e-246:
		tmp = t_1
	elif z <= -4.3e-302:
		tmp = x * math.log(y)
	elif z <= 3.3e+78:
		tmp = t_1
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(t) - y)
	tmp = 0.0
	if (z <= -3.2e+111)
		tmp = Float64(-z);
	elseif (z <= -2.3e-246)
		tmp = t_1;
	elseif (z <= -4.3e-302)
		tmp = Float64(x * log(y));
	elseif (z <= 3.3e+78)
		tmp = t_1;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(t) - y;
	tmp = 0.0;
	if (z <= -3.2e+111)
		tmp = -z;
	elseif (z <= -2.3e-246)
		tmp = t_1;
	elseif (z <= -4.3e-302)
		tmp = x * log(y);
	elseif (z <= 3.3e+78)
		tmp = t_1;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[z, -3.2e+111], (-z), If[LessEqual[z, -2.3e-246], t$95$1, If[LessEqual[z, -4.3e-302], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.3e+78], t$95$1, (-z)]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.2000000000000001e111 or 3.3e78 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 86.3%

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

      1. mul-1-neg 86.3%

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

   4. Simplified 86.3%

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

   if -3.2000000000000001e111 < z < -2.2999999999999998e-246 or -4.3000000000000002e-302 < z < 3.3e78

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. add-log-exp 19.7%

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

      3. sum-log 19.7%

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

      4. associate--l- 19.7%

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

      5. exp-diff 18.5%

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

      6. *-commutative 18.5%

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

      7. exp-to-pow 18.5%

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

   3. Applied egg-rr 18.5%

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

   4. Taylor expanded in x around 0 18.0%

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

   5. Taylor expanded in z around 0 17.8%

      \[\leadsto \color{blue}{\log \left(\frac{t}{e^{y}}\right)} \]
   6. Step-by-step derivation

      1. log-div 17.8%

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

      2. rem-log-exp 61.0%

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

   7. Simplified 61.0%

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

   if -2.2999999999999998e-246 < z < -4.3000000000000002e-302

   1. Initial program 99.6%

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

   2. Taylor expanded in z around 0 99.6%

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

   3. Taylor expanded in x around inf 75.0%

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

      1. *-commutative 75.0%

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

   5. Simplified 75.0%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+111}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{-246}:\\ \;\;\;\;\log t - y\\ \mathbf{elif}\;z \leq -4.3 \cdot 10^{-302}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+78}:\\ \;\;\;\;\log t - y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 5: 82.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+94} \lor \neg \left(x \leq 8.5 \cdot 10^{+212}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -5.6e+94) (not (<= x 8.5e+212)))
   (* x (log y))
   (- (log t) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -5.6e+94) || !(x <= 8.5e+212)) {
		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 <= (-5.6d+94)) .or. (.not. (x <= 8.5d+212))) 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 <= -5.6e+94) || !(x <= 8.5e+212)) {
		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 <= -5.6e+94) or not (x <= 8.5e+212):
		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 <= -5.6e+94) || !(x <= 8.5e+212))
		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 <= -5.6e+94) || ~((x <= 8.5e+212)))
		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, -5.6e+94], N[Not[LessEqual[x, 8.5e+212]], $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 < -5.59999999999999997e94 or 8.49999999999999979e212 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in z around 0 86.4%

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

   3. Taylor expanded in x around inf 77.8%

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

      1. *-commutative 77.8%

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

   5. Simplified 77.8%

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

   if -5.59999999999999997e94 < x < 8.49999999999999979e212

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 91.3%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+94} \lor \neg \left(x \leq 8.5 \cdot 10^{+212}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]

Alternative 6: 88.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+94} \lor \neg \left(x \leq 7 \cdot 10^{+140}\right):\\ \;\;\;\;x \cdot \log y - z\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.55e+94) (not (<= x 7e+140)))
   (- (* x (log y)) z)
   (- (log t) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.55e+94) || !(x <= 7e+140)) {
		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 <= (-1.55d+94)) .or. (.not. (x <= 7d+140))) 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 <= -1.55e+94) || !(x <= 7e+140)) {
		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 <= -1.55e+94) or not (x <= 7e+140):
		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 <= -1.55e+94) || !(x <= 7e+140))
		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 <= -1.55e+94) || ~((x <= 7e+140)))
		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, -1.55e+94], N[Not[LessEqual[x, 7e+140]], $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 < -1.54999999999999996e94 or 6.99999999999999978e140 < x

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. sub-neg 99.7%

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

      3. associate--l+ 99.7%

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

      4. add-cube-cbrt 98.5%

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

      5. associate-*r* 98.5%

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

      6. fma-def 98.5%

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

      7. pow2 98.5%

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

   3. Applied egg-rr 98.5%

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

   4. Taylor expanded in z around inf 87.2%

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

      1. neg-mul-1 87.2%

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

   6. Simplified 87.2%

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

      1. fma-udef 87.2%

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

      2. unsub-neg 87.2%

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

      3. associate-*r* 87.1%

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

      4. unpow2 87.1%

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

      5. add-cube-cbrt 88.3%

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

      6. *-commutative 88.3%

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

   8. Applied egg-rr 88.3%

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

   if -1.54999999999999996e94 < x < 6.99999999999999978e140

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 94.1%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+94} \lor \neg \left(x \leq 7 \cdot 10^{+140}\right):\\ \;\;\;\;x \cdot \log y - z\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]

Alternative 7: 48.5% accurate, 34.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+111} \lor \neg \left(z \leq 3 \cdot 10^{+78}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -3.3e+111) (not (<= z 3e+78))) (- z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.3e+111) || !(z <= 3e+78)) {
		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 ((z <= (-3.3d+111)) .or. (.not. (z <= 3d+78))) 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 ((z <= -3.3e+111) || !(z <= 3e+78)) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -3.3e+111) or not (z <= 3e+78):
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -3.3e+111) || !(z <= 3e+78))
		tmp = Float64(-z);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -3.3e+111) || ~((z <= 3e+78)))
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -3.3e+111], N[Not[LessEqual[z, 3e+78]], $MachinePrecision]], (-z), (-y)]

Derivation

1. Split input into 2 regimes

2. if z < -3.3000000000000001e111 or 2.99999999999999982e78 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 86.3%

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

      1. mul-1-neg 86.3%

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

   4. Simplified 86.3%

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

   if -3.3000000000000001e111 < z < 2.99999999999999982e78

   1. Initial program 99.8%

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

   2. Taylor expanded in y around inf 44.8%

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

      1. neg-mul-1 44.8%

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

   4. Simplified 44.8%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+111} \lor \neg \left(z \leq 3 \cdot 10^{+78}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 8: 30.3% accurate, 104.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in y around inf 31.5%

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

   1. neg-mul-1 31.5%

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

4. Simplified 31.5%

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

5. Final simplification 31.5%

   \[\leadsto -y \]

Reproduce

herbie shell --seed 2023318 
(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)))