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

Percentage Accurate: 99.9% → 99.9%

Time: 11.8s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. associate--l- N/A

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

   2. associate-+l- N/A

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

   3. *-commutative N/A

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

   4. fmm-def N/A

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

   5. fma-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\log y, \color{blue}{x}, \left(\mathsf{neg}\left(\left(\left(y + z\right) - \log t\right)\right)\right)\right) \]

   6. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(y\right), x, \left(\mathsf{neg}\left(\left(\left(y + z\right) - \log t\right)\right)\right)\right) \]

   7. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(y\right), x, \mathsf{neg.f64}\left(\left(\left(y + z\right) - \log t\right)\right)\right) \]

   8. associate-+r- N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(y\right), x, \mathsf{neg.f64}\left(\left(y + \left(z - \log t\right)\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(y\right), x, \mathsf{neg.f64}\left(\mathsf{+.f64}\left(y, \left(z - \log t\right)\right)\right)\right) \]

   10. --lowering--.f64 N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(y\right), x, \mathsf{neg.f64}\left(\mathsf{+.f64}\left(y, \mathsf{\_.f64}\left(z, \log t\right)\right)\right)\right) \]

   11. log-lowering-log.f64 99.9%

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(y\right), x, \mathsf{neg.f64}\left(\mathsf{+.f64}\left(y, \mathsf{\_.f64}\left(z, \mathsf{log.f64}\left(t\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 88.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\log y, x, 0 - y\right)\\ \mathbf{if}\;x \leq -3.45 \cdot 10^{+93}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+114}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (log y) x (- 0.0 y))))
   (if (<= x -3.45e+93) t_1 (if (<= x 4.8e+114) (- (- (log t) z) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(log(y), x, (0.0 - y));
	double tmp;
	if (x <= -3.45e+93) {
		tmp = t_1;
	} else if (x <= 4.8e+114) {
		tmp = (log(t) - z) - y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(log(y), x, Float64(0.0 - y))
	tmp = 0.0
	if (x <= -3.45e+93)
		tmp = t_1;
	elseif (x <= 4.8e+114)
		tmp = Float64(Float64(log(t) - z) - y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x + N[(0.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.45e+93], t$95$1, If[LessEqual[x, 4.8e+114], N[(N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision] - y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.4499999999999998e93 or 4.8e114 < 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. associate--l- N/A

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

      2. associate-+l- N/A

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

      3. *-commutative N/A

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

      4. fmm-def N/A

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

      5. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\log y, \color{blue}{x}, \left(\mathsf{neg}\left(\left(\left(y + z\right) - \log t\right)\right)\right)\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(y\right), x, \left(\mathsf{neg}\left(\left(\left(y + z\right) - \log t\right)\right)\right)\right) \]

      7. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(y\right), x, \mathsf{neg.f64}\left(\left(\left(y + z\right) - \log t\right)\right)\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(y\right), x, \mathsf{neg.f64}\left(\left(y + \left(z - \log t\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(y\right), x, \mathsf{neg.f64}\left(\mathsf{+.f64}\left(y, \left(z - \log t\right)\right)\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(y\right), x, \mathsf{neg.f64}\left(\mathsf{+.f64}\left(y, \mathsf{\_.f64}\left(z, \log t\right)\right)\right)\right) \]

      11. log-lowering-log.f64 99.7%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(y\right), x, \mathsf{neg.f64}\left(\mathsf{+.f64}\left(y, \mathsf{\_.f64}\left(z, \mathsf{log.f64}\left(t\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(y\right), x, \mathsf{neg.f64}\left(\color{blue}{y}\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 89.0%

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

   if -3.4499999999999998e93 < x < 4.8e114

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t + \left(\mathsf{neg}\left(z\right)\right)\right), \color{blue}{y}\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t - z\right), y\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\log t, z\right), y\right) \]

      7. log-lowering-log.f64 96.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{log.f64}\left(t\right), z\right), y\right) \]

   5. Simplified 96.4%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.45 \cdot 10^{+93}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, 0 - y\right)\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+114}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, 0 - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{+90}:\\ \;\;\;\;\log t + \left(\log y \cdot x - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, 0 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.35e+90)
   (+ (log t) (- (* (log y) x) z))
   (fma (log y) x (- 0.0 y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.35e+90) {
		tmp = log(t) + ((log(y) * x) - z);
	} else {
		tmp = fma(log(y), x, (0.0 - y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.35e+90)
		tmp = Float64(log(t) + Float64(Float64(log(y) * x) - z));
	else
		tmp = fma(log(y), x, Float64(0.0 - y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.35e+90], N[(N[Log[t], $MachinePrecision] + N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x + N[(0.0 - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.35e90

   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

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(x \cdot \log y - z\right)}, \mathsf{log.f64}\left(t\right)\right) \]
   4. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot \log y\right), z\right), \mathsf{log.f64}\left(\color{blue}{t}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \log y\right), z\right), \mathsf{log.f64}\left(t\right)\right) \]

      3. log-lowering-log.f64 94.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), z\right), \mathsf{log.f64}\left(t\right)\right) \]

   5. Simplified 94.9%

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

   if 1.35e90 < 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. associate--l- N/A

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

      2. associate-+l- N/A

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

      3. *-commutative N/A

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

      4. fmm-def N/A

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

      5. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\log y, \color{blue}{x}, \left(\mathsf{neg}\left(\left(\left(y + z\right) - \log t\right)\right)\right)\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(y\right), x, \left(\mathsf{neg}\left(\left(\left(y + z\right) - \log t\right)\right)\right)\right) \]

      7. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(y\right), x, \mathsf{neg.f64}\left(\left(\left(y + z\right) - \log t\right)\right)\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(y\right), x, \mathsf{neg.f64}\left(\left(y + \left(z - \log t\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(y\right), x, \mathsf{neg.f64}\left(\mathsf{+.f64}\left(y, \left(z - \log t\right)\right)\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(y\right), x, \mathsf{neg.f64}\left(\mathsf{+.f64}\left(y, \mathsf{\_.f64}\left(z, \log t\right)\right)\right)\right) \]

      11. log-lowering-log.f64 99.9%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(y\right), x, \mathsf{neg.f64}\left(\mathsf{+.f64}\left(y, \mathsf{\_.f64}\left(z, \mathsf{log.f64}\left(t\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{log.f64}\left(y\right), x, \mathsf{neg.f64}\left(\color{blue}{y}\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 92.3%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{+90}:\\ \;\;\;\;\log t + \left(\log y \cdot x - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, 0 - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 5: 82.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -7 \cdot 10^{+159}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+164}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -7e+159) t_1 (if (<= x 7.2e+164) (- (- (log t) z) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if (x <= -7e+159) {
		tmp = t_1;
	} else if (x <= 7.2e+164) {
		tmp = (log(t) - z) - y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(y) * x
    if (x <= (-7d+159)) then
        tmp = t_1
    else if (x <= 7.2d+164) then
        tmp = (log(t) - z) - y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) * x;
	double tmp;
	if (x <= -7e+159) {
		tmp = t_1;
	} else if (x <= 7.2e+164) {
		tmp = (Math.log(t) - z) - y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	tmp = 0
	if x <= -7e+159:
		tmp = t_1
	elif x <= 7.2e+164:
		tmp = (math.log(t) - z) - y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (x <= -7e+159)
		tmp = t_1;
	elseif (x <= 7.2e+164)
		tmp = Float64(Float64(log(t) - z) - y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	tmp = 0.0;
	if (x <= -7e+159)
		tmp = t_1;
	elseif (x <= 7.2e+164)
		tmp = (log(t) - z) - y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -7e+159], t$95$1, If[LessEqual[x, 7.2e+164], N[(N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision] - y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.9999999999999999e159 or 7.19999999999999981e164 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\log y}\right) \]

      2. log-lowering-log.f64 75.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right) \]

   5. Simplified 75.0%

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

   if -6.9999999999999999e159 < x < 7.19999999999999981e164

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t + \left(\mathsf{neg}\left(z\right)\right)\right), \color{blue}{y}\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t - z\right), y\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\log t, z\right), y\right) \]

      7. log-lowering-log.f64 91.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{log.f64}\left(t\right), z\right), y\right) \]

   5. Simplified 91.9%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+159}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+164}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(0 - y\right) - z\\ \mathbf{if}\;z \leq -550:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-26}:\\ \;\;\;\;\log t - y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (- 0.0 y) z)))
   (if (<= z -550.0) t_1 (if (<= z 5.5e-26) (- (log t) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (0.0 - y) - z;
	double tmp;
	if (z <= -550.0) {
		tmp = t_1;
	} else if (z <= 5.5e-26) {
		tmp = log(t) - y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (0.0d0 - y) - z
    if (z <= (-550.0d0)) then
        tmp = t_1
    else if (z <= 5.5d-26) then
        tmp = log(t) - y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (0.0 - y) - z;
	double tmp;
	if (z <= -550.0) {
		tmp = t_1;
	} else if (z <= 5.5e-26) {
		tmp = Math.log(t) - y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (0.0 - y) - z
	tmp = 0
	if z <= -550.0:
		tmp = t_1
	elif z <= 5.5e-26:
		tmp = math.log(t) - y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(0.0 - y) - z)
	tmp = 0.0
	if (z <= -550.0)
		tmp = t_1;
	elseif (z <= 5.5e-26)
		tmp = Float64(log(t) - y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (0.0 - y) - z;
	tmp = 0.0;
	if (z <= -550.0)
		tmp = t_1;
	elseif (z <= 5.5e-26)
		tmp = log(t) - y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(0.0 - y), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[z, -550.0], t$95$1, If[LessEqual[z, 5.5e-26], N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -550 or 5.5000000000000005e-26 < 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. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t + \left(\mathsf{neg}\left(z\right)\right)\right), \color{blue}{y}\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t - z\right), y\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\log t, z\right), y\right) \]

      7. log-lowering-log.f64 82.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{log.f64}\left(t\right), z\right), y\right) \]

   5. Simplified 82.6%

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

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot z\right)}, y\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), y\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(0 - z\right), y\right) \]

      3. --lowering--.f64 81.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), y\right) \]

   8. Simplified 81.2%

      \[\leadsto \color{blue}{\left(0 - z\right)} - y \]

   if -550 < z < 5.5000000000000005e-26

   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. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t + \left(\mathsf{neg}\left(z\right)\right)\right), \color{blue}{y}\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t - z\right), y\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\log t, z\right), y\right) \]

      7. log-lowering-log.f64 67.1%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{log.f64}\left(t\right), z\right), y\right) \]

   5. Simplified 67.1%

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

   6. Taylor expanded in z around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\log t, \color{blue}{y}\right) \]

      2. log-lowering-log.f64 66.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log.f64}\left(t\right), y\right) \]

   8. Simplified 66.5%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -550:\\ \;\;\;\;\left(0 - y\right) - z\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-26}:\\ \;\;\;\;\log t - y\\ \mathbf{else}:\\ \;\;\;\;\left(0 - y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -9 \cdot 10^{+159}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+165}:\\ \;\;\;\;\left(0 - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -9e+159) t_1 (if (<= x 6.2e+165) (- (- 0.0 y) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if (x <= -9e+159) {
		tmp = t_1;
	} else if (x <= 6.2e+165) {
		tmp = (0.0 - y) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(y) * x
    if (x <= (-9d+159)) then
        tmp = t_1
    else if (x <= 6.2d+165) then
        tmp = (0.0d0 - y) - z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) * x;
	double tmp;
	if (x <= -9e+159) {
		tmp = t_1;
	} else if (x <= 6.2e+165) {
		tmp = (0.0 - y) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	tmp = 0
	if x <= -9e+159:
		tmp = t_1
	elif x <= 6.2e+165:
		tmp = (0.0 - y) - z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (x <= -9e+159)
		tmp = t_1;
	elseif (x <= 6.2e+165)
		tmp = Float64(Float64(0.0 - y) - z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	tmp = 0.0;
	if (x <= -9e+159)
		tmp = t_1;
	elseif (x <= 6.2e+165)
		tmp = (0.0 - y) - z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -9e+159], t$95$1, If[LessEqual[x, 6.2e+165], N[(N[(0.0 - y), $MachinePrecision] - z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.00000000000000053e159 or 6.2000000000000003e165 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\log y}\right) \]

      2. log-lowering-log.f64 75.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right) \]

   5. Simplified 75.0%

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

   if -9.00000000000000053e159 < x < 6.2000000000000003e165

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t + \left(\mathsf{neg}\left(z\right)\right)\right), \color{blue}{y}\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t - z\right), y\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\log t, z\right), y\right) \]

      7. log-lowering-log.f64 91.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{log.f64}\left(t\right), z\right), y\right) \]

   5. Simplified 91.9%

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

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot z\right)}, y\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), y\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(0 - z\right), y\right) \]

      3. --lowering--.f64 71.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), y\right) \]

   8. Simplified 71.2%

      \[\leadsto \color{blue}{\left(0 - z\right)} - y \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+159}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+165}:\\ \;\;\;\;\left(0 - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{+14}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;\left(0 - y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 5.5e+14) (- (log t) z) (- (- 0.0 y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 5.5e+14) {
		tmp = log(t) - z;
	} else {
		tmp = (0.0 - y) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 5.5d+14) then
        tmp = log(t) - z
    else
        tmp = (0.0d0 - y) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 5.5e+14) {
		tmp = Math.log(t) - z;
	} else {
		tmp = (0.0 - y) - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 5.5e+14:
		tmp = math.log(t) - z
	else:
		tmp = (0.0 - y) - z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 5.5e+14)
		tmp = Float64(log(t) - z);
	else
		tmp = Float64(Float64(0.0 - y) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 5.5e+14)
		tmp = log(t) - z;
	else
		tmp = (0.0 - y) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 5.5e+14], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], N[(N[(0.0 - y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.5e14

   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. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t + \left(\mathsf{neg}\left(z\right)\right)\right), \color{blue}{y}\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t - z\right), y\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\log t, z\right), y\right) \]

      7. log-lowering-log.f64 72.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{log.f64}\left(t\right), z\right), y\right) \]

   5. Simplified 72.6%

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

   6. Taylor expanded in y around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\log t, \color{blue}{z}\right) \]

      2. log-lowering-log.f64 71.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log.f64}\left(t\right), z\right) \]

   8. Simplified 71.7%

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

   if 5.5e14 < 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. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t + \left(\mathsf{neg}\left(z\right)\right)\right), \color{blue}{y}\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t - z\right), y\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\log t, z\right), y\right) \]

      7. log-lowering-log.f64 78.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{log.f64}\left(t\right), z\right), y\right) \]

   5. Simplified 78.0%

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

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot z\right)}, y\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), y\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(0 - z\right), y\right) \]

      3. --lowering--.f64 78.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), y\right) \]

   8. Simplified 78.0%

      \[\leadsto \color{blue}{\left(0 - z\right)} - y \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{+14}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;\left(0 - y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{-171}:\\ \;\;\;\;\log t\\ \mathbf{else}:\\ \;\;\;\;\left(0 - y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 1e-171) (log t) (- (- 0.0 y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1e-171) {
		tmp = log(t);
	} else {
		tmp = (0.0 - y) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 1d-171) then
        tmp = log(t)
    else
        tmp = (0.0d0 - y) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1e-171) {
		tmp = Math.log(t);
	} else {
		tmp = (0.0 - y) - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 1e-171:
		tmp = math.log(t)
	else:
		tmp = (0.0 - y) - z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1e-171)
		tmp = log(t);
	else
		tmp = Float64(Float64(0.0 - y) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 1e-171)
		tmp = log(t);
	else
		tmp = (0.0 - y) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1e-171], N[Log[t], $MachinePrecision], N[(N[(0.0 - y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.9999999999999998e-172

   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. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t + \left(\mathsf{neg}\left(z\right)\right)\right), \color{blue}{y}\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t - z\right), y\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\log t, z\right), y\right) \]

      7. log-lowering-log.f64 73.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{log.f64}\left(t\right), z\right), y\right) \]

   5. Simplified 73.4%

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

   6. Taylor expanded in z around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\log t, \color{blue}{y}\right) \]

      2. log-lowering-log.f64 41.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log.f64}\left(t\right), y\right) \]

   8. Simplified 41.6%

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

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\log t} \]
   10. Step-by-step derivation

       1. log-lowering-log.f64 41.6%

          \[\leadsto \mathsf{log.f64}\left(t\right) \]

   11. Simplified 41.6%

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

   if 9.9999999999999998e-172 < 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. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t + \left(\mathsf{neg}\left(z\right)\right)\right), \color{blue}{y}\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t - z\right), y\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\log t, z\right), y\right) \]

      7. log-lowering-log.f64 75.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{log.f64}\left(t\right), z\right), y\right) \]

   5. Simplified 75.9%

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

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot z\right)}, y\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), y\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(0 - z\right), y\right) \]

      3. --lowering--.f64 68.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), y\right) \]

   8. Simplified 68.3%

      \[\leadsto \color{blue}{\left(0 - z\right)} - y \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-171}:\\ \;\;\;\;\log t\\ \mathbf{else}:\\ \;\;\;\;\left(0 - y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.6% accurate, 26.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{+93}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;0 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 2.2e+93) (- 0.0 z) (- 0.0 y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.2e+93) {
		tmp = 0.0 - z;
	} else {
		tmp = 0.0 - 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 <= 2.2d+93) then
        tmp = 0.0d0 - z
    else
        tmp = 0.0d0 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.2e+93) {
		tmp = 0.0 - z;
	} else {
		tmp = 0.0 - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 2.2e+93:
		tmp = 0.0 - z
	else:
		tmp = 0.0 - y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.2e+93)
		tmp = Float64(0.0 - z);
	else
		tmp = Float64(0.0 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 2.2e+93)
		tmp = 0.0 - z;
	else
		tmp = 0.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 2.2e+93], N[(0.0 - z), $MachinePrecision], N[(0.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.20000000000000021e93

   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

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

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{z} \]

      3. --lowering--.f64 42.8%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{z}\right) \]

   5. Simplified 42.8%

      \[\leadsto \color{blue}{0 - z} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 42.8%

         \[\leadsto \mathsf{neg.f64}\left(z\right) \]

   7. Applied egg-rr 42.8%

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

   if 2.20000000000000021e93 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(y\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{y} \]

      3. --lowering--.f64 70.7%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{y}\right) \]

   5. Simplified 70.7%

      \[\leadsto \color{blue}{0 - y} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(y\right) \]

      2. neg-lowering-neg.f64 70.7%

         \[\leadsto \mathsf{neg.f64}\left(y\right) \]

   7. Applied egg-rr 70.7%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{+93}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;0 - y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.6% accurate, 41.8× speedup

Math

\[\begin{array}{l} \\ \left(0 - y\right) - z \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (- 0.0 y) z))

C

double code(double x, double y, double z, double t) {
	return (0.0 - 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 = (0.0d0 - y) - z
end function

Java

public static double code(double x, double y, double z, double t) {
	return (0.0 - y) - z;
}

Python

def code(x, y, z, t):
	return (0.0 - y) - z

Julia

function code(x, y, z, t)
	return Float64(Float64(0.0 - y) - z)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (0.0 - y) - z;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(0.0 - y), $MachinePrecision] - z), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate--r+ N/A

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

   3. unsub-neg N/A

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

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\log t + \left(\mathsf{neg}\left(z\right)\right)\right), \color{blue}{y}\right) \]

   5. unsub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\log t - z\right), y\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\log t, z\right), y\right) \]

   7. log-lowering-log.f64 75.3%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{log.f64}\left(t\right), z\right), y\right) \]

5. Simplified 75.3%

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

6. Taylor expanded in z around inf

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot z\right)}, y\right) \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), y\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(0 - z\right), y\right) \]

   3. --lowering--.f64 59.7%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), y\right) \]

8. Simplified 59.7%

   \[\leadsto \color{blue}{\left(0 - z\right)} - y \]

9. Final simplification 59.7%

   \[\leadsto \left(0 - y\right) - z \]
10. Add Preprocessing

Alternative 12: 29.3% accurate, 69.7× speedup

Math

\[\begin{array}{l} \\ 0 - y \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- 0.0 y))

C

double code(double x, double y, double z, double t) {
	return 0.0 - y;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 0.0d0 - y
end function

Java

public static double code(double x, double y, double z, double t) {
	return 0.0 - y;
}

Python

def code(x, y, z, t):
	return 0.0 - y

Julia

function code(x, y, z, t)
	return Float64(0.0 - y)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 0.0 - y;
end

Wolfram

code[x_, y_, z_, t_] := N[(0.0 - y), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(y\right) \]

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{y} \]

   3. --lowering--.f64 32.5%

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{y}\right) \]

5. Simplified 32.5%

   \[\leadsto \color{blue}{0 - y} \]
6. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{neg}\left(y\right) \]

   2. neg-lowering-neg.f64 32.5%

      \[\leadsto \mathsf{neg.f64}\left(y\right) \]

7. Applied egg-rr 32.5%

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

8. Final simplification 32.5%

   \[\leadsto 0 - y \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024158 
(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)))