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

Percentage Accurate: 99.9% → 99.9%

Time: 12.4s

Alternatives: 15

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: 92.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - y\\ t_2 := \mathsf{fma}\left(x, \log y, -z\right)\\ \mathbf{if}\;t\_1 \leq -5000000000000:\\ \;\;\;\;y \cdot \left(\frac{t\_2}{y} + -1\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-13}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) y)) (t_2 (fma x (log y) (- z))))
   (if (<= t_1 -5000000000000.0)
     (* y (+ (/ t_2 y) -1.0))
     (if (<= t_1 4e-13) (- (log t) (+ y z)) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - y;
	double t_2 = fma(x, log(y), -z);
	double tmp;
	if (t_1 <= -5000000000000.0) {
		tmp = y * ((t_2 / y) + -1.0);
	} else if (t_1 <= 4e-13) {
		tmp = log(t) - (y + z);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - y)
	t_2 = fma(x, log(y), Float64(-z))
	tmp = 0.0
	if (t_1 <= -5000000000000.0)
		tmp = Float64(y * Float64(Float64(t_2 / y) + -1.0));
	elseif (t_1 <= 4e-13)
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[Log[y], $MachinePrecision] + (-z)), $MachinePrecision]}, If[LessEqual[t$95$1, -5000000000000.0], N[(y * N[(N[(t$95$2 / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-13], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -5e12

   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-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. flip3-+ N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 99.7

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

   7. Simplified 99.7%

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

   8. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate--r+ N/A

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

   10. Simplified 91.9%

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

   if -5e12 < (-.f64 (*.f64 x (log.f64 y)) y) < 4.0000000000000001e-13

   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. lower--.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-+.f64 99.9

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

   5. Simplified 99.9%

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

   if 4.0000000000000001e-13 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.7%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   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. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 99.7

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

   5. Simplified 99.7%

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

   6. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.6

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

   8. Simplified 99.6%

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

Alternative 3: 92.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - y\\ \mathbf{if}\;t\_1 \leq -5000000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\log y, \frac{x}{y}, -1\right), y, -z\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-13}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) y)))
   (if (<= t_1 -5000000000000.0)
     (fma (fma (log y) (/ x y) -1.0) y (- z))
     (if (<= t_1 4e-13) (- (log t) (+ y z)) (fma x (log y) (- z))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - y;
	double tmp;
	if (t_1 <= -5000000000000.0) {
		tmp = fma(fma(log(y), (x / y), -1.0), y, -z);
	} else if (t_1 <= 4e-13) {
		tmp = log(t) - (y + z);
	} else {
		tmp = fma(x, log(y), -z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - y)
	tmp = 0.0
	if (t_1 <= -5000000000000.0)
		tmp = fma(fma(log(y), Float64(x / y), -1.0), y, Float64(-z));
	elseif (t_1 <= 4e-13)
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = fma(x, log(y), Float64(-z));
	end
	return 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, -5000000000000.0], N[(N[(N[Log[y], $MachinePrecision] * N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision] * y + (-z)), $MachinePrecision], If[LessEqual[t$95$1, 4e-13], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], N[(x * N[Log[y], $MachinePrecision] + (-z)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -5e12

   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

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. distribute-frac-neg N/A

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

      10. remove-double-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-log.f64 92.0

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

   5. Simplified 92.0%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-neg.f64 N/A

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

      8. associate-+l+ N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lower-fma.f64 92.0

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

   7. Applied egg-rr 91.9%

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

   8. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\log y, \frac{x}{y}, -1\right), y, \color{blue}{-1 \cdot z}\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 91.9

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

   10. Simplified 91.9%

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

   if -5e12 < (-.f64 (*.f64 x (log.f64 y)) y) < 4.0000000000000001e-13

   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. lower--.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-+.f64 99.9

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

   5. Simplified 99.9%

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

   if 4.0000000000000001e-13 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.7%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   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. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 99.7

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

   5. Simplified 99.7%

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

   6. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.6

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

   8. Simplified 99.6%

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

Alternative 4: 69.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \log y - y\right) - z\\ \mathbf{if}\;t\_1 \leq -20000000000:\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-13}:\\ \;\;\;\;\log t\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (- (* x (log y)) y) z)))
   (if (<= t_1 -20000000000.0) (- (- z) y) (if (<= t_1 4e-13) (log t) (- z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((x * log(y)) - y) - z;
	double tmp;
	if (t_1 <= -20000000000.0) {
		tmp = -z - y;
	} else if (t_1 <= 4e-13) {
		tmp = log(t);
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((x * log(y)) - y) - z
    if (t_1 <= (-20000000000.0d0)) then
        tmp = -z - y
    else if (t_1 <= 4d-13) then
        tmp = log(t)
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((x * Math.log(y)) - y) - z;
	double tmp;
	if (t_1 <= -20000000000.0) {
		tmp = -z - y;
	} else if (t_1 <= 4e-13) {
		tmp = Math.log(t);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((x * math.log(y)) - y) - z
	tmp = 0
	if t_1 <= -20000000000.0:
		tmp = -z - y
	elif t_1 <= 4e-13:
		tmp = math.log(t)
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(x * log(y)) - y) - z)
	tmp = 0.0
	if (t_1 <= -20000000000.0)
		tmp = Float64(Float64(-z) - y);
	elseif (t_1 <= 4e-13)
		tmp = log(t);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((x * log(y)) - y) - z;
	tmp = 0.0;
	if (t_1 <= -20000000000.0)
		tmp = -z - y;
	elseif (t_1 <= 4e-13)
		tmp = log(t);
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$1, -20000000000.0], N[((-z) - y), $MachinePrecision], If[LessEqual[t$95$1, 4e-13], N[Log[t], $MachinePrecision], (-z)]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) < -2e10

   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-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. flip3-+ N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 99.4

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

   7. Simplified 99.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \left(y + z\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

      4. sub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-neg.f64 74.7

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

   10. Simplified 74.7%

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

   if -2e10 < (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) < 4.0000000000000001e-13

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 97.6

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

   5. Simplified 97.6%

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

   6. Taylor expanded in z around 0

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

      1. lower-log.f64 97.0

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

   8. Simplified 97.0%

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

   if 4.0000000000000001e-13 < (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z)

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 47.5

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

   5. Simplified 47.5%

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

Alternative 5: 78.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t\_1 - y\\ \mathbf{if}\;t\_2 \leq -30000000000:\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{elif}\;t\_2 \leq 10^{+111}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- t_1 y)))
   (if (<= t_2 -30000000000.0)
     (- (- z) y)
     (if (<= t_2 1e+111) (- (log t) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = t_1 - y;
	double tmp;
	if (t_2 <= -30000000000.0) {
		tmp = -z - y;
	} else if (t_2 <= 1e+111) {
		tmp = log(t) - 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) :: t_2
    real(8) :: tmp
    t_1 = x * log(y)
    t_2 = t_1 - y
    if (t_2 <= (-30000000000.0d0)) then
        tmp = -z - y
    else if (t_2 <= 1d+111) then
        tmp = log(t) - 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 = x * Math.log(y);
	double t_2 = t_1 - y;
	double tmp;
	if (t_2 <= -30000000000.0) {
		tmp = -z - y;
	} else if (t_2 <= 1e+111) {
		tmp = Math.log(t) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = t_1 - y
	tmp = 0
	if t_2 <= -30000000000.0:
		tmp = -z - y
	elif t_2 <= 1e+111:
		tmp = math.log(t) - z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(t_1 - y)
	tmp = 0.0
	if (t_2 <= -30000000000.0)
		tmp = Float64(Float64(-z) - y);
	elseif (t_2 <= 1e+111)
		tmp = Float64(log(t) - z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	t_2 = t_1 - y;
	tmp = 0.0;
	if (t_2 <= -30000000000.0)
		tmp = -z - y;
	elseif (t_2 <= 1e+111)
		tmp = log(t) - z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - y), $MachinePrecision]}, If[LessEqual[t$95$2, -30000000000.0], N[((-z) - y), $MachinePrecision], If[LessEqual[t$95$2, 1e+111], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -3e10

   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-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. flip3-+ N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 99.4

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

   7. Simplified 99.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \left(y + z\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

      4. sub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-neg.f64 72.7

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

   10. Simplified 72.7%

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

   if -3e10 < (-.f64 (*.f64 x (log.f64 y)) y) < 9.99999999999999957e110

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 95.2

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

   5. Simplified 95.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. lift--.f64 95.2

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

   7. Applied egg-rr 95.2%

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

   if 9.99999999999999957e110 < (-.f64 (*.f64 x (log.f64 y)) y)

   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. lower-*.f64 N/A

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

      2. lower-log.f64 90.5

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

   5. Simplified 90.5%

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

Alternative 6: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 420000000:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, \log t - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{\mathsf{fma}\left(x, \log y, -z\right)}{y} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 420000000.0)
   (fma x (log y) (- (log t) z))
   (* y (+ (/ (fma x (log y) (- z)) y) -1.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 420000000.0) {
		tmp = fma(x, log(y), (log(t) - z));
	} else {
		tmp = y * ((fma(x, log(y), -z) / y) + -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 420000000.0)
		tmp = fma(x, log(y), Float64(log(t) - z));
	else
		tmp = Float64(y * Float64(Float64(fma(x, log(y), Float64(-z)) / y) + -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 420000000.0], N[(x * N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(N[(x * N[Log[y], $MachinePrecision] + (-z)), $MachinePrecision] / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.2e8

   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. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 99.2

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

   5. Simplified 99.2%

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

   if 4.2e8 < y

   1. Initial program 100.0%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. flip3-+ N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 99.4

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

   7. Simplified 99.4%

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

   8. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate--r+ N/A

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

   10. Simplified 99.6%

       \[\leadsto \color{blue}{y \cdot \left(\frac{\mathsf{fma}\left(x, \log y, -z\right)}{y} + -1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 99.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\frac{1}{x \cdot \log y - \left(y + z\right)}}\\ \mathbf{if}\;x \leq -19000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.98:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 1.0 (/ 1.0 (- (* x (log y)) (+ y z))))))
   (if (<= x -19000.0) t_1 (if (<= x 0.98) (- (log t) (+ y z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / (1.0 / ((x * log(y)) - (y + z)));
	double tmp;
	if (x <= -19000.0) {
		tmp = t_1;
	} else if (x <= 0.98) {
		tmp = log(t) - (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 = 1.0d0 / (1.0d0 / ((x * log(y)) - (y + z)))
    if (x <= (-19000.0d0)) then
        tmp = t_1
    else if (x <= 0.98d0) then
        tmp = log(t) - (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 = 1.0 / (1.0 / ((x * Math.log(y)) - (y + z)));
	double tmp;
	if (x <= -19000.0) {
		tmp = t_1;
	} else if (x <= 0.98) {
		tmp = Math.log(t) - (y + z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 1.0 / (1.0 / ((x * math.log(y)) - (y + z)))
	tmp = 0
	if x <= -19000.0:
		tmp = t_1
	elif x <= 0.98:
		tmp = math.log(t) - (y + z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(1.0 / Float64(1.0 / Float64(Float64(x * log(y)) - Float64(y + z))))
	tmp = 0.0
	if (x <= -19000.0)
		tmp = t_1;
	elseif (x <= 0.98)
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 / (1.0 / ((x * log(y)) - (y + z)));
	tmp = 0.0;
	if (x <= -19000.0)
		tmp = t_1;
	elseif (x <= 0.98)
		tmp = log(t) - (y + z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 / N[(1.0 / N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -19000.0], t$95$1, If[LessEqual[x, 0.98], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -19000 or 0.97999999999999998 < x

   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-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. flip3-+ N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 99.6

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

   7. Simplified 99.6%

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

   if -19000 < x < 0.97999999999999998

   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. lower--.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-+.f64 100.0

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

   5. Simplified 100.0%

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

Alternative 8: 89.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - y\\ \mathbf{if}\;x \leq -3.05 \cdot 10^{+68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+25}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) y)))
   (if (<= x -3.05e+68) t_1 (if (<= x 2.6e+25) (- (log t) (+ y z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - y;
	double tmp;
	if (x <= -3.05e+68) {
		tmp = t_1;
	} else if (x <= 2.6e+25) {
		tmp = log(t) - (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 = (x * log(y)) - y
    if (x <= (-3.05d+68)) then
        tmp = t_1
    else if (x <= 2.6d+25) then
        tmp = log(t) - (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 = (x * Math.log(y)) - y;
	double tmp;
	if (x <= -3.05e+68) {
		tmp = t_1;
	} else if (x <= 2.6e+25) {
		tmp = Math.log(t) - (y + z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - y
	tmp = 0
	if x <= -3.05e+68:
		tmp = t_1
	elif x <= 2.6e+25:
		tmp = math.log(t) - (y + z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - y)
	tmp = 0.0
	if (x <= -3.05e+68)
		tmp = t_1;
	elseif (x <= 2.6e+25)
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * log(y)) - y;
	tmp = 0.0;
	if (x <= -3.05e+68)
		tmp = t_1;
	elseif (x <= 2.6e+25)
		tmp = log(t) - (y + z);
	else
		tmp = t_1;
	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[x, -3.05e+68], t$95$1, If[LessEqual[x, 2.6e+25], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.05e68 or 2.5999999999999998e25 < x

   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-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. flip3-+ N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 99.6

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

   7. Simplified 99.6%

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

   8. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 87.8

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

   10. Simplified 87.8%

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

   if -3.05e68 < x < 2.5999999999999998e25

   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. lower--.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-+.f64 98.2

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

   5. Simplified 98.2%

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

Alternative 9: 82.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -2.8 \cdot 10^{+111}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+32}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -2.8e+111) t_1 (if (<= x 5e+32) (- (log t) (+ y z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -2.8e+111) {
		tmp = t_1;
	} else if (x <= 5e+32) {
		tmp = log(t) - (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 = x * log(y)
    if (x <= (-2.8d+111)) then
        tmp = t_1
    else if (x <= 5d+32) then
        tmp = log(t) - (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 = x * Math.log(y);
	double tmp;
	if (x <= -2.8e+111) {
		tmp = t_1;
	} else if (x <= 5e+32) {
		tmp = Math.log(t) - (y + z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -2.8e+111:
		tmp = t_1
	elif x <= 5e+32:
		tmp = math.log(t) - (y + z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -2.8e+111)
		tmp = t_1;
	elseif (x <= 5e+32)
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (x <= -2.8e+111)
		tmp = t_1;
	elseif (x <= 5e+32)
		tmp = log(t) - (y + z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.8e+111], t$95$1, If[LessEqual[x, 5e+32], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.7999999999999999e111 or 4.9999999999999997e32 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 69.5

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

   5. Simplified 69.5%

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

   if -2.7999999999999999e111 < x < 4.9999999999999997e32

   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. lower--.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-+.f64 96.1

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

   5. Simplified 96.1%

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

Alternative 10: 70.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-z\right) - y\\ \mathbf{if}\;z \leq -8.2 \cdot 10^{-24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-10}:\\ \;\;\;\;\log t - y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (- z) y)))
   (if (<= z -8.2e-24) t_1 (if (<= z 4.8e-10) (- (log t) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -z - y;
	double tmp;
	if (z <= -8.2e-24) {
		tmp = t_1;
	} else if (z <= 4.8e-10) {
		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 = -z - y
    if (z <= (-8.2d-24)) then
        tmp = t_1
    else if (z <= 4.8d-10) 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 = -z - y;
	double tmp;
	if (z <= -8.2e-24) {
		tmp = t_1;
	} else if (z <= 4.8e-10) {
		tmp = Math.log(t) - y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -z - y
	tmp = 0
	if z <= -8.2e-24:
		tmp = t_1
	elif z <= 4.8e-10:
		tmp = math.log(t) - y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-z) - y)
	tmp = 0.0
	if (z <= -8.2e-24)
		tmp = t_1;
	elseif (z <= 4.8e-10)
		tmp = Float64(log(t) - y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -z - y;
	tmp = 0.0;
	if (z <= -8.2e-24)
		tmp = t_1;
	elseif (z <= 4.8e-10)
		tmp = log(t) - y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-z) - y), $MachinePrecision]}, If[LessEqual[z, -8.2e-24], t$95$1, If[LessEqual[z, 4.8e-10], N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.20000000000000029e-24 or 4.8e-10 < z

   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-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. flip3-+ N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 99.6

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \left(y + z\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

      4. sub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-neg.f64 73.6

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

   10. Simplified 73.6%

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

   if -8.20000000000000029e-24 < z < 4.8e-10

   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

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 68.8

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

   5. Simplified 68.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 68.8

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

   7. Applied egg-rr 68.8%

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

Alternative 11: 70.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 280:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 280.0) (- (log t) z) (- (- z) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 280.0) {
		tmp = log(t) - z;
	} else {
		tmp = -z - 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 <= 280.0d0) then
        tmp = log(t) - z
    else
        tmp = -z - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 280.0) {
		tmp = Math.log(t) - z;
	} else {
		tmp = -z - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 280.0:
		tmp = math.log(t) - z
	else:
		tmp = -z - y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 280.0)
		tmp = Float64(log(t) - z);
	else
		tmp = Float64(Float64(-z) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 280.0)
		tmp = log(t) - z;
	else
		tmp = -z - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 280.0], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], N[((-z) - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 280

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 60.7

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

   5. Simplified 60.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. lift--.f64 60.7

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

   7. Applied egg-rr 60.7%

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

   if 280 < y

   1. Initial program 100.0%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. flip3-+ N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 99.4

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

   7. Simplified 99.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \left(y + z\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

      4. sub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-neg.f64 81.6

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

   10. Simplified 81.6%

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

Alternative 12: 48.8% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+17}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+45}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.25e+17) (- z) (if (<= z 1.7e+45) (- y) (- z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.25e+17) {
		tmp = -z;
	} else if (z <= 1.7e+45) {
		tmp = -y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1.25d+17)) then
        tmp = -z
    else if (z <= 1.7d+45) then
        tmp = -y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.25e+17) {
		tmp = -z;
	} else if (z <= 1.7e+45) {
		tmp = -y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.25e+17:
		tmp = -z
	elif z <= 1.7e+45:
		tmp = -y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.25e+17)
		tmp = Float64(-z);
	elseif (z <= 1.7e+45)
		tmp = Float64(-y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.25e+17)
		tmp = -z;
	elseif (z <= 1.7e+45)
		tmp = -y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.25e+17], (-z), If[LessEqual[z, 1.7e+45], (-y), (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -1.25e17 or 1.7e45 < 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 63.4

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

   5. Simplified 63.4%

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

   if -1.25e17 < z < 1.7e45

   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

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

      2. lower-neg.f64 42.4

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

   5. Simplified 42.4%

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

Alternative 13: 58.8% accurate, 35.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[((-z) - y), $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. lift-log.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-log.f64 N/A

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

   6. flip3-+ N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-log.f64 85.8

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

7. Simplified 85.8%

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

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{-1 \cdot \left(y + z\right)} \]
9. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-neg-in N/A

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

   4. sub-neg N/A

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

   5. lower--.f64 N/A

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

   6. lower-neg.f64 57.5

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

10. Simplified 57.5%

    \[\leadsto \color{blue}{\left(-z\right) - y} \]
11. Add Preprocessing

Alternative 14: 29.7% accurate, 71.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 29.6

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

5. Simplified 29.6%

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

Alternative 15: 2.2% accurate, 215.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := y

Derivation

1. Initial program 99.9%

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

   1. lift-log.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-log.f64 N/A

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

   6. flip3-+ N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in y around inf

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

   1. lower-/.f64 29.6

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

7. Simplified 29.6%

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

   1. frac-2neg N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(y\right)}}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{1}}{\mathsf{neg}\left(y\right)}} \]

   3. lift-neg.f64 N/A

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\mathsf{neg}\left(y\right)}}} \]

   4. inv-pow N/A

      \[\leadsto \frac{1}{\color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{-1}}} \]

   5. pow-flip N/A

      \[\leadsto \color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{\left(\mathsf{neg}\left(-1\right)\right)}} \]

   6. metadata-eval N/A

      \[\leadsto {\left(\mathsf{neg}\left(y\right)\right)}^{\color{blue}{1}} \]

   7. metadata-eval N/A

      \[\leadsto {\left(\mathsf{neg}\left(y\right)\right)}^{\color{blue}{\left(3 - 2\right)}} \]

   8. pow-div N/A

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3}}{{\left(\mathsf{neg}\left(y\right)\right)}^{2}}} \]

   9. sqr-pow N/A

      \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(y\right)\right)}^{\left(\frac{3}{2}\right)}}}{{\left(\mathsf{neg}\left(y\right)\right)}^{2}} \]

   10. unpow-prod-down N/A

       \[\leadsto \frac{\color{blue}{{\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{{\left(\mathsf{neg}\left(y\right)\right)}^{2}} \]

   11. lift-neg.f64 N/A

       \[\leadsto \frac{{\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{{\left(\mathsf{neg}\left(y\right)\right)}^{2}} \]

   12. lift-neg.f64 N/A

       \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)}^{\left(\frac{3}{2}\right)}}{{\left(\mathsf{neg}\left(y\right)\right)}^{2}} \]

   13. sqr-neg N/A

       \[\leadsto \frac{{\color{blue}{\left(y \cdot y\right)}}^{\left(\frac{3}{2}\right)}}{{\left(\mathsf{neg}\left(y\right)\right)}^{2}} \]

   14. unpow-prod-down N/A

       \[\leadsto \frac{\color{blue}{{y}^{\left(\frac{3}{2}\right)} \cdot {y}^{\left(\frac{3}{2}\right)}}}{{\left(\mathsf{neg}\left(y\right)\right)}^{2}} \]

   15. sqr-pow N/A

       \[\leadsto \frac{\color{blue}{{y}^{3}}}{{\left(\mathsf{neg}\left(y\right)\right)}^{2}} \]

   16. pow2 N/A

       \[\leadsto \frac{{y}^{3}}{\color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)}} \]

   17. lift-neg.f64 N/A

       \[\leadsto \frac{{y}^{3}}{\color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \left(\mathsf{neg}\left(y\right)\right)} \]

   18. lift-neg.f64 N/A

       \[\leadsto \frac{{y}^{3}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}} \]

   19. sqr-neg N/A

       \[\leadsto \frac{{y}^{3}}{\color{blue}{y \cdot y}} \]

   20. pow2 N/A

       \[\leadsto \frac{{y}^{3}}{\color{blue}{{y}^{2}}} \]

   21. pow-div N/A

       \[\leadsto \color{blue}{{y}^{\left(3 - 2\right)}} \]

   22. metadata-eval N/A

       \[\leadsto {y}^{\color{blue}{1}} \]

   23. unpow1 2.0

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

9. Applied egg-rr 2.0%

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

Reproduce

herbie shell --seed 2024207 
(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)))