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

Percentage Accurate: 99.9% → 99.9%

Time: 9.5s

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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * N[Log[y], $MachinePrecision] + N[((-y) - z), $MachinePrecision]), $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. 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-def 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. Final simplification 99.9%

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

Alternative 2: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - y\\ \mathbf{if}\;t_1 \leq -5000 \lor \neg \left(t_1 \leq 10^{-8}\right):\\ \;\;\;\;t_1 - z\\ \mathbf{else}:\\ \;\;\;\;\log \left(t \cdot {y}^{x}\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) y)))
   (if (or (<= t_1 -5000.0) (not (<= t_1 1e-8)))
     (- t_1 z)
     (- (log (* t (pow y x))) z))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - y;
	double tmp;
	if ((t_1 <= -5000.0) || !(t_1 <= 1e-8)) {
		tmp = t_1 - z;
	} else {
		tmp = log((t * pow(y, x))) - 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)) - y
    if ((t_1 <= (-5000.0d0)) .or. (.not. (t_1 <= 1d-8))) then
        tmp = t_1 - z
    else
        tmp = log((t * (y ** x))) - 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)) - y;
	double tmp;
	if ((t_1 <= -5000.0) || !(t_1 <= 1e-8)) {
		tmp = t_1 - z;
	} else {
		tmp = Math.log((t * Math.pow(y, x))) - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - y
	tmp = 0
	if (t_1 <= -5000.0) or not (t_1 <= 1e-8):
		tmp = t_1 - z
	else:
		tmp = math.log((t * math.pow(y, x))) - z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - y)
	tmp = 0.0
	if ((t_1 <= -5000.0) || !(t_1 <= 1e-8))
		tmp = Float64(t_1 - z);
	else
		tmp = Float64(log(Float64(t * (y ^ x))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * log(y)) - y;
	tmp = 0.0;
	if ((t_1 <= -5000.0) || ~((t_1 <= 1e-8)))
		tmp = t_1 - z;
	else
		tmp = log((t * (y ^ x))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5000.0], N[Not[LessEqual[t$95$1, 1e-8]], $MachinePrecision]], N[(t$95$1 - z), $MachinePrecision], N[(N[Log[N[(t * N[Power[y, x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -5e3 or 1e-8 < (-.f64 (*.f64 x (log.f64 y)) y)

   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%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 99.2%

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

   if -5e3 < (-.f64 (*.f64 x (log.f64 y)) y) < 1e-8

   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-def 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 0 99.8%

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

      1. remove-double-neg 99.8%

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

      2. mul-1-neg 99.8%

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

      3. log-rec 99.8%

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

      4. log-rec 99.8%

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

      5. mul-1-neg 99.8%

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

      6. remove-double-neg 99.8%

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

      7. log-pow 99.8%

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

      8. log-prod 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log y - y \leq -5000 \lor \neg \left(x \cdot \log y - y \leq 10^{-8}\right):\\ \;\;\;\;\left(x \cdot \log y - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\log \left(t \cdot {y}^{x}\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - y\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+31} \lor \neg \left(t_1 \leq 10^{-8}\right):\\ \;\;\;\;t_1 - z\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) y)))
   (if (or (<= t_1 -5e+31) (not (<= t_1 1e-8)))
     (- t_1 z)
     (- (log t) (+ y z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - y;
	double tmp;
	if ((t_1 <= -5e+31) || !(t_1 <= 1e-8)) {
		tmp = t_1 - 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) :: t_1
    real(8) :: tmp
    t_1 = (x * log(y)) - y
    if ((t_1 <= (-5d+31)) .or. (.not. (t_1 <= 1d-8))) then
        tmp = t_1 - 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 t_1 = (x * Math.log(y)) - y;
	double tmp;
	if ((t_1 <= -5e+31) || !(t_1 <= 1e-8)) {
		tmp = t_1 - z;
	} else {
		tmp = Math.log(t) - (y + z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - y
	tmp = 0
	if (t_1 <= -5e+31) or not (t_1 <= 1e-8):
		tmp = t_1 - z
	else:
		tmp = math.log(t) - (y + z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - y)
	tmp = 0.0
	if ((t_1 <= -5e+31) || !(t_1 <= 1e-8))
		tmp = Float64(t_1 - z);
	else
		tmp = Float64(log(t) - Float64(y + z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * log(y)) - y;
	tmp = 0.0;
	if ((t_1 <= -5e+31) || ~((t_1 <= 1e-8)))
		tmp = t_1 - z;
	else
		tmp = log(t) - (y + z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e+31], N[Not[LessEqual[t$95$1, 1e-8]], $MachinePrecision]], N[(t$95$1 - z), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -5.00000000000000027e31 or 1e-8 < (-.f64 (*.f64 x (log.f64 y)) y)

   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%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 99.6%

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

   if -5.00000000000000027e31 < (-.f64 (*.f64 x (log.f64 y)) y) < 1e-8

   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-def 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 98.4%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log y - y \leq -5 \cdot 10^{+31} \lor \neg \left(x \cdot \log y - y \leq 10^{-8}\right):\\ \;\;\;\;\left(x \cdot \log y - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 5: 59.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+89} \lor \neg \left(z \leq 20000000000\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\log t - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -5.2e+89) (not (<= z 20000000000.0))) (- z) (- (log t) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -5.2e+89) || !(z <= 20000000000.0)) {
		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 <= (-5.2d+89)) .or. (.not. (z <= 20000000000.0d0))) 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 <= -5.2e+89) || !(z <= 20000000000.0)) {
		tmp = -z;
	} else {
		tmp = Math.log(t) - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -5.2e+89) or not (z <= 20000000000.0):
		tmp = -z
	else:
		tmp = math.log(t) - y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -5.2e+89) || !(z <= 20000000000.0))
		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 <= -5.2e+89) || ~((z <= 20000000000.0)))
		tmp = -z;
	else
		tmp = log(t) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -5.2e+89], N[Not[LessEqual[z, 20000000000.0]], $MachinePrecision]], (-z), N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.2000000000000001e89 or 2e10 < z

   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-def 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 z around inf 60.2%

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

      1. neg-mul-1 60.2%

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

   7. Simplified 60.2%

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

   8. Taylor expanded in z around inf 60.2%

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

      1. mul-1-neg 60.2%

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

   10. Simplified 60.2%

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

   if -5.2000000000000001e89 < z < 2e10

   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-def 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 z around 0 96.6%

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

      1. fma-neg 96.6%

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

   7. Simplified 96.6%

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

   8. Taylor expanded in x around 0 63.2%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+89} \lor \neg \left(z \leq 20000000000\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\log t - y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 40.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+18} \lor \neg \left(z \leq 1.7 \cdot 10^{-23}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.5e+18) (not (<= z 1.7e-23))) (- z) (log t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.5e+18) || !(z <= 1.7e-23)) {
		tmp = -z;
	} else {
		tmp = log(t);
	}
	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 <= (-2.5d+18)) .or. (.not. (z <= 1.7d-23))) then
        tmp = -z
    else
        tmp = log(t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.5e+18) || !(z <= 1.7e-23)) {
		tmp = -z;
	} else {
		tmp = Math.log(t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2.5e+18) or not (z <= 1.7e-23):
		tmp = -z
	else:
		tmp = math.log(t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.5e+18) || !(z <= 1.7e-23))
		tmp = Float64(-z);
	else
		tmp = log(t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -2.5e+18) || ~((z <= 1.7e-23)))
		tmp = -z;
	else
		tmp = log(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2.5e+18], N[Not[LessEqual[z, 1.7e-23]], $MachinePrecision]], (-z), N[Log[t], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.5e18 or 1.7e-23 < z

   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-def 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 z around inf 54.9%

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

      1. neg-mul-1 54.9%

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

   7. Simplified 54.9%

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

   8. Taylor expanded in z around inf 55.0%

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

      1. mul-1-neg 55.0%

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

   10. Simplified 55.0%

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

   if -2.5e18 < z < 1.7e-23

   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-def 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 z around inf 26.7%

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

      1. neg-mul-1 26.7%

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

   7. Simplified 26.7%

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

   8. Taylor expanded in z around 0 25.8%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+18} \lor \neg \left(z \leq 1.7 \cdot 10^{-23}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\log t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+117}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;\log t - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 1e+117) (- (log t) z) (- (log t) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1e+117) {
		tmp = log(t) - 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 (y <= 1d+117) then
        tmp = log(t) - 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 (y <= 1e+117) {
		tmp = Math.log(t) - z;
	} else {
		tmp = Math.log(t) - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 1e+117:
		tmp = math.log(t) - z
	else:
		tmp = math.log(t) - y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1e+117)
		tmp = Float64(log(t) - z);
	else
		tmp = Float64(log(t) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 1e+117)
		tmp = log(t) - z;
	else
		tmp = log(t) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1e+117], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.00000000000000005e117

   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-def 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 z around inf 55.9%

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

      1. neg-mul-1 55.9%

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

   7. Simplified 55.9%

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

   8. Taylor expanded in z around 0 55.9%

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

      1. mul-1-neg 55.9%

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

      2. sub-neg 55.9%

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

   10. Simplified 55.9%

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

   if 1.00000000000000005e117 < 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-def 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 0 83.7%

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

      1. fma-neg 83.7%

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

   7. Simplified 83.7%

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

   8. Taylor expanded in x around 0 71.0%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+117}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;\log t - y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.2% 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. 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-def 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 x around 0 71.9%

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

6. Final simplification 71.9%

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

Alternative 9: 29.3% accurate, 104.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return -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 = -z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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-def 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 z around inf 41.4%

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

   1. neg-mul-1 41.4%

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

7. Simplified 41.4%

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

8. Taylor expanded in z around inf 29.8%

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

   1. mul-1-neg 29.8%

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

10. Simplified 29.8%

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

11. Final simplification 29.8%

    \[\leadsto -z \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(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)))