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

Percentage Accurate: 99.9% → 99.9%

Time: 9.2s

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: 68.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - y\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+113}:\\ \;\;\;\;\log t - y\\ \mathbf{elif}\;t\_1 \leq 10^{+19}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) y)))
   (if (<= t_1 -5e+113)
     (- (log t) y)
     (if (<= t_1 1e+19) (- (log t) z) (* (log y) x)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - y;
	double tmp;
	if (t_1 <= -5e+113) {
		tmp = log(t) - y;
	} else if (t_1 <= 1e+19) {
		tmp = log(t) - z;
	} else {
		tmp = log(y) * x;
	}
	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+113)) then
        tmp = log(t) - y
    else if (t_1 <= 1d+19) then
        tmp = log(t) - z
    else
        tmp = log(y) * x
    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+113) {
		tmp = Math.log(t) - y;
	} else if (t_1 <= 1e+19) {
		tmp = Math.log(t) - z;
	} else {
		tmp = Math.log(y) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - y
	tmp = 0
	if t_1 <= -5e+113:
		tmp = math.log(t) - y
	elif t_1 <= 1e+19:
		tmp = math.log(t) - z
	else:
		tmp = math.log(y) * x
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - y)
	tmp = 0.0
	if (t_1 <= -5e+113)
		tmp = Float64(log(t) - y);
	elseif (t_1 <= 1e+19)
		tmp = Float64(log(t) - z);
	else
		tmp = Float64(log(y) * x);
	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+113)
		tmp = log(t) - y;
	elseif (t_1 <= 1e+19)
		tmp = log(t) - z;
	else
		tmp = log(y) * x;
	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[LessEqual[t$95$1, -5e+113], N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision], If[LessEqual[t$95$1, 1e+19], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -5e113

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}}} \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}}}} \]

      6. flip3-+ N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 78.7

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

   7. Applied rewrites 78.7%

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

   8. Taylor expanded in x around 0

      \[\leadsto \log t - y \]
   9. Step-by-step derivation

   10. Applied rewrites 54.3%

       \[\leadsto \log t - y \]

   if -5e113 < (-.f64 (*.f64 x (log.f64 y)) y) < 1e19

   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 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 88.8

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

   5. Applied rewrites 88.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \log t - z \]
   7. Step-by-step derivation

   8. Applied rewrites 77.2%

      \[\leadsto \log t - z \]

   if 1e19 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 79.7

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

   7. Applied rewrites 79.7%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log y - y \leq -5 \cdot 10^{+113}:\\ \;\;\;\;\log t - y\\ \mathbf{elif}\;x \cdot \log y - y \leq 10^{+19}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.013 \lor \neg \left(z \leq 6.2 \cdot 10^{+18}\right):\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -0.013) (not (<= z 6.2e+18)))
   (- (- (log t) y) z)
   (- (fma (log y) x (log t)) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -0.013) || !(z <= 6.2e+18)) {
		tmp = (log(t) - y) - z;
	} else {
		tmp = fma(log(y), x, log(t)) - y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -0.013) || !(z <= 6.2e+18))
		tmp = Float64(Float64(log(t) - y) - z);
	else
		tmp = Float64(fma(log(y), x, log(t)) - y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -0.013], N[Not[LessEqual[z, 6.2e+18]], $MachinePrecision]], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * x + N[Log[t], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.0129999999999999994 or 6.2e18 < z

   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

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 86.6

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

   5. Applied rewrites 86.6%

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

   if -0.0129999999999999994 < z < 6.2e18

   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

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.013 \lor \neg \left(z \leq 6.2 \cdot 10^{+18}\right):\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t\right) - y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 3.8e+47) (- (fma (log y) x (log t)) z) (- (- (log t) y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.8e+47) {
		tmp = fma(log(y), x, log(t)) - z;
	} else {
		tmp = (log(t) - y) - z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3.8e+47)
		tmp = Float64(fma(log(y), x, log(t)) - z);
	else
		tmp = Float64(Float64(log(t) - y) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 3.8e+47], N[(N[(N[Log[y], $MachinePrecision] * x + N[Log[t], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.8000000000000003e47

   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

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 95.9

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

   5. Applied rewrites 95.9%

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

   if 3.8000000000000003e47 < 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

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 86.9

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

   5. Applied rewrites 86.9%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 48.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \lor \neg \left(z \leq 4.8 \cdot 10^{+28}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{-1}{y}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.1) (not (<= z 4.8e+28))) (- z) (pow (/ -1.0 y) -1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.1) || !(z <= 4.8e+28)) {
		tmp = -z;
	} else {
		tmp = pow((-1.0 / y), -1.0);
	}
	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 <= (-1.1d0)) .or. (.not. (z <= 4.8d+28))) then
        tmp = -z
    else
        tmp = ((-1.0d0) / y) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.1) || !(z <= 4.8e+28)) {
		tmp = -z;
	} else {
		tmp = Math.pow((-1.0 / y), -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.1) or not (z <= 4.8e+28):
		tmp = -z
	else:
		tmp = math.pow((-1.0 / y), -1.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.1) || !(z <= 4.8e+28))
		tmp = Float64(-z);
	else
		tmp = Float64(-1.0 / y) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.1) || ~((z <= 4.8e+28)))
		tmp = -z;
	else
		tmp = (-1.0 / y) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.1], N[Not[LessEqual[z, 4.8e+28]], $MachinePrecision]], (-z), N[Power[N[(-1.0 / y), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.1000000000000001 or 4.79999999999999962e28 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. lower-neg.f64 67.2

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

   5. Applied rewrites 67.2%

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

   if -1.1000000000000001 < z < 4.79999999999999962e28

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}}} \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}}}} \]

      6. flip3-+ N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 39.8

         \[\leadsto \frac{1}{\color{blue}{\frac{-1}{y}}} \]

   7. Applied rewrites 39.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{-1}{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \lor \neg \left(z \leq 4.8 \cdot 10^{+28}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{-1}{y}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.36 \cdot 10^{+108} \lor \neg \left(x \leq 1.35 \cdot 10^{+235}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.36e+108) (not (<= x 1.35e+235)))
   (* (log y) x)
   (- (- (log t) y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.36e+108) || !(x <= 1.35e+235)) {
		tmp = log(y) * x;
	} 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 <= (-1.36d+108)) .or. (.not. (x <= 1.35d+235))) then
        tmp = log(y) * x
    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 <= -1.36e+108) || !(x <= 1.35e+235)) {
		tmp = Math.log(y) * x;
	} else {
		tmp = (Math.log(t) - y) - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.36e+108) or not (x <= 1.35e+235):
		tmp = math.log(y) * x
	else:
		tmp = (math.log(t) - y) - z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.36e+108) || !(x <= 1.35e+235))
		tmp = Float64(log(y) * x);
	else
		tmp = Float64(Float64(log(t) - y) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.36e+108) || ~((x <= 1.35e+235)))
		tmp = log(y) * x;
	else
		tmp = (log(t) - y) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.36e+108], N[Not[LessEqual[x, 1.35e+235]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3599999999999999e108 or 1.3499999999999999e235 < x

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 96.3%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 77.5

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

   7. Applied rewrites 77.5%

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

   if -1.3599999999999999e108 < x < 1.3499999999999999e235

   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

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 87.9

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

   5. Applied rewrites 87.9%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.36 \cdot 10^{+108} \lor \neg \left(x \leq 1.35 \cdot 10^{+235}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{+112}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{-1}{y}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 6.5e+112) (- (log t) z) (pow (/ -1.0 y) -1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 6.5e+112) {
		tmp = log(t) - z;
	} else {
		tmp = pow((-1.0 / y), -1.0);
	}
	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 <= 6.5d+112) then
        tmp = log(t) - z
    else
        tmp = ((-1.0d0) / y) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 6.5e+112) {
		tmp = Math.log(t) - z;
	} else {
		tmp = Math.pow((-1.0 / y), -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 6.5e+112:
		tmp = math.log(t) - z
	else:
		tmp = math.pow((-1.0 / y), -1.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 6.5e+112)
		tmp = Float64(log(t) - z);
	else
		tmp = Float64(-1.0 / y) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 6.5e+112)
		tmp = log(t) - z;
	else
		tmp = (-1.0 / y) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 6.5e+112], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], N[Power[N[(-1.0 / y), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.4999999999999998e112

   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

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 92.5

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

   5. Applied rewrites 92.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \log t - z \]
   7. Step-by-step derivation

   8. Applied rewrites 59.2%

      \[\leadsto \log t - z \]

   if 6.4999999999999998e112 < y

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}}} \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}}}} \]

      6. flip3-+ N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 68.7

         \[\leadsto \frac{1}{\color{blue}{\frac{-1}{y}}} \]

   7. Applied rewrites 68.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{-1}{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{+112}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{-1}{y}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.027:\\ \;\;\;\;\log t - z\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+28}:\\ \;\;\;\;\log t - y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -0.027) (- (log t) z) (if (<= z 6.8e+28) (- (log t) y) (- z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -0.027) {
		tmp = log(t) - z;
	} else if (z <= 6.8e+28) {
		tmp = log(t) - 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) :: tmp
    if (z <= (-0.027d0)) then
        tmp = log(t) - z
    else if (z <= 6.8d+28) then
        tmp = log(t) - y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -0.027) {
		tmp = Math.log(t) - z;
	} else if (z <= 6.8e+28) {
		tmp = Math.log(t) - y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -0.027:
		tmp = math.log(t) - z
	elif z <= 6.8e+28:
		tmp = math.log(t) - y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -0.027)
		tmp = Float64(log(t) - z);
	elseif (z <= 6.8e+28)
		tmp = Float64(log(t) - y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -0.027)
		tmp = log(t) - z;
	elseif (z <= 6.8e+28)
		tmp = log(t) - y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -0.027], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], If[LessEqual[z, 6.8e+28], N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision], (-z)]]

Derivation

1. Split input into 3 regimes

2. if z < -0.0269999999999999997

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 85.8

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

   5. Applied rewrites 85.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \log t - z \]
   7. Step-by-step derivation

   8. Applied rewrites 71.8%

      \[\leadsto \log t - z \]

   if -0.0269999999999999997 < z < 6.8e28

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}}} \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(\left(x \cdot \log y - y\right) - z\right)}^{3} + {\log t}^{3}}{\left(\left(x \cdot \log y - y\right) - z\right) \cdot \left(\left(x \cdot \log y - y\right) - z\right) + \left(\log t \cdot \log t - \left(\left(x \cdot \log y - y\right) - z\right) \cdot \log t\right)}}}} \]

      6. flip3-+ N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 99.1

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in x around 0

      \[\leadsto \log t - y \]
   9. Step-by-step derivation

   10. Applied rewrites 59.8%

       \[\leadsto \log t - y \]

   if 6.8e28 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. lower-neg.f64 66.0

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

   5. Applied rewrites 66.0%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.027:\\ \;\;\;\;\log t - z\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+28}:\\ \;\;\;\;\log t - y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 30.3% accurate, 71.7× 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. Add Preprocessing

3. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

   2. lower-neg.f64 35.4

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

5. Applied rewrites 35.4%

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

6. Final simplification 35.4%

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

Reproduce

herbie shell --seed 2024324 
(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)))