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

Percentage Accurate: 99.9% → 99.9%

Time: 13.4s

Alternatives: 14

Speedup: 1.0×

• could not determine a ground truth

  1. x = 3.131994195891661e-222
  2. y = 2.3680707764782997e+63
  3. z = 4.1693899127050236e+63
  4. t = 3.7974310193297695e-180

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-+l- 99.8%

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

   2. sub-neg 99.8%

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

   3. associate--l+ 99.8%

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

   4. fma-define 99.9%

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

   5. sub-neg 99.9%

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

   6. distribute-neg-in 99.9%

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

   7. sub-neg 99.9%

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

   8. associate-+r+ 99.9%

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

   9. +-commutative 99.9%

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

   10. distribute-neg-in 99.9%

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

   11. remove-double-neg 99.9%

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

   12. unsub-neg 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - y\\ \mathbf{if}\;t\_1 \leq -2.6 \cdot 10^{+18} \lor \neg \left(t\_1 \leq 2 \cdot 10^{-32}\right):\\ \;\;\;\;t\_1 - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) y)))
   (if (or (<= t_1 -2.6e+18) (not (<= t_1 2e-32)))
     (- t_1 z)
     (- (- (log t) z) y))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - y;
	double tmp;
	if ((t_1 <= -2.6e+18) || !(t_1 <= 2e-32)) {
		tmp = t_1 - z;
	} else {
		tmp = (log(t) - 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) :: t_1
    real(8) :: tmp
    t_1 = (x * log(y)) - y
    if ((t_1 <= (-2.6d+18)) .or. (.not. (t_1 <= 2d-32))) then
        tmp = t_1 - z
    else
        tmp = (log(t) - z) - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * Math.log(y)) - y;
	double tmp;
	if ((t_1 <= -2.6e+18) || !(t_1 <= 2e-32)) {
		tmp = t_1 - z;
	} else {
		tmp = (Math.log(t) - z) - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - y
	tmp = 0
	if (t_1 <= -2.6e+18) or not (t_1 <= 2e-32):
		tmp = t_1 - z
	else:
		tmp = (math.log(t) - z) - y
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - y)
	tmp = 0.0
	if ((t_1 <= -2.6e+18) || !(t_1 <= 2e-32))
		tmp = Float64(t_1 - z);
	else
		tmp = Float64(Float64(log(t) - z) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * log(y)) - y;
	tmp = 0.0;
	if ((t_1 <= -2.6e+18) || ~((t_1 <= 2e-32)))
		tmp = t_1 - z;
	else
		tmp = (log(t) - z) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2.6e+18], N[Not[LessEqual[t$95$1, 2e-32]], $MachinePrecision]], N[(t$95$1 - z), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision] - y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -2.6e18 or 2.00000000000000011e-32 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.8%

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

      1. associate-+l- 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 99.5%

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

   if -2.6e18 < (-.f64 (*.f64 x (log.f64 y)) y) < 2.00000000000000011e-32

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

   3. Simplified 100.0%

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

      1. add-cube-cbrt 99.8%

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

      2. pow2 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 99.5%

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

      1. neg-mul-1 99.5%

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

   9. Simplified 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log y - y \leq -2.6 \cdot 10^{+18} \lor \neg \left(x \cdot \log y - y \leq 2 \cdot 10^{-32}\right):\\ \;\;\;\;\left(x \cdot \log y - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - y\\ \mathbf{if}\;t\_1 \leq -2.6 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -\left(y + z\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-32}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \mathbf{else}:\\ \;\;\;\;t\_1 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) y)))
   (if (<= t_1 -2.6e+18)
     (fma (log y) x (- (+ y z)))
     (if (<= t_1 2e-32) (- (- (log t) z) y) (- t_1 z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - y;
	double tmp;
	if (t_1 <= -2.6e+18) {
		tmp = fma(log(y), x, -(y + z));
	} else if (t_1 <= 2e-32) {
		tmp = (log(t) - z) - y;
	} else {
		tmp = t_1 - z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - y)
	tmp = 0.0
	if (t_1 <= -2.6e+18)
		tmp = fma(log(y), x, Float64(-Float64(y + z)));
	elseif (t_1 <= 2e-32)
		tmp = Float64(Float64(log(t) - z) - y);
	else
		tmp = Float64(t_1 - 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, -2.6e+18], N[(N[Log[y], $MachinePrecision] * x + (-N[(y + z), $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$1, 2e-32], N[(N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision] - y), $MachinePrecision], N[(t$95$1 - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -2.6e18

   1. Initial program 99.8%

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

      1. associate-+l- 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 99.8%

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

      1. sub-neg 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. fma-define 99.9%

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

      6. *-un-lft-identity 99.9%

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

      7. fma-define 99.9%

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

      8. fma-neg 99.9%

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

      9. *-un-lft-identity 99.9%

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

   7. Applied egg-rr 99.9%

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

   if -2.6e18 < (-.f64 (*.f64 x (log.f64 y)) y) < 2.00000000000000011e-32

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

   3. Simplified 100.0%

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

      1. add-cube-cbrt 99.8%

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

      2. pow2 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 99.5%

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

      1. neg-mul-1 99.5%

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

   9. Simplified 99.5%

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

   if 2.00000000000000011e-32 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.6%

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

      1. associate-+l- 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in z around inf 98.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log y - y \leq -2.6 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -\left(y + z\right)\right)\\ \mathbf{elif}\;x \cdot \log y - y \leq 2 \cdot 10^{-32}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log y - y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0017:\\ \;\;\;\;\left(\log t + x \cdot \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -\left(y + z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 0.0017)
   (- (+ (log t) (* x (log y))) z)
   (fma (log y) x (- (+ y z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 0.0017) {
		tmp = (log(t) + (x * log(y))) - z;
	} else {
		tmp = fma(log(y), x, -(y + z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 0.0017)
		tmp = Float64(Float64(log(t) + Float64(x * log(y))) - z);
	else
		tmp = fma(log(y), x, Float64(-Float64(y + z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 0.0017], N[(N[(N[Log[t], $MachinePrecision] + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x + (-N[(y + z), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.00169999999999999991

   1. Initial program 99.8%

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

      1. associate-+l- 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 99.5%

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

   if 0.00169999999999999991 < y

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 98.0%

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

      1. sub-neg 98.0%

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

      2. sub-neg 98.0%

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

      3. associate-+l+ 98.0%

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

      4. *-commutative 98.0%

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

      5. fma-define 98.0%

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

      6. *-un-lft-identity 98.0%

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

      7. fma-define 98.0%

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

      8. fma-neg 98.0%

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

      9. *-un-lft-identity 98.0%

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

   7. Applied egg-rr 98.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0017:\\ \;\;\;\;\left(\log t + x \cdot \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -\left(y + z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-+l- 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 7: 82.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+122} \lor \neg \left(x \leq 1.8 \cdot 10^{+16} \lor \neg \left(x \leq 1.34 \cdot 10^{+73}\right) \land x \leq 6.6 \cdot 10^{+172}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.4e+122)
         (not (or (<= x 1.8e+16) (and (not (<= x 1.34e+73)) (<= x 6.6e+172)))))
   (* x (log y))
   (- (log t) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.4e+122) || !((x <= 1.8e+16) || (!(x <= 1.34e+73) && (x <= 6.6e+172)))) {
		tmp = x * log(y);
	} else {
		tmp = log(t) - (y + z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-1.4d+122)) .or. (.not. (x <= 1.8d+16) .or. (.not. (x <= 1.34d+73)) .and. (x <= 6.6d+172))) then
        tmp = x * log(y)
    else
        tmp = log(t) - (y + z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.4e+122) || !((x <= 1.8e+16) || (!(x <= 1.34e+73) && (x <= 6.6e+172)))) {
		tmp = x * Math.log(y);
	} else {
		tmp = Math.log(t) - (y + z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.4e+122) or not ((x <= 1.8e+16) or (not (x <= 1.34e+73) and (x <= 6.6e+172))):
		tmp = x * math.log(y)
	else:
		tmp = math.log(t) - (y + z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.4e+122) || !((x <= 1.8e+16) || (!(x <= 1.34e+73) && (x <= 6.6e+172))))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(log(t) - Float64(y + z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.4e+122) || ~(((x <= 1.8e+16) || (~((x <= 1.34e+73)) && (x <= 6.6e+172)))))
		tmp = x * log(y);
	else
		tmp = log(t) - (y + z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.4e+122], N[Not[Or[LessEqual[x, 1.8e+16], And[N[Not[LessEqual[x, 1.34e+73]], $MachinePrecision], LessEqual[x, 6.6e+172]]]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4e122 or 1.8e16 < x < 1.34e73 or 6.59999999999999965e172 < x

   1. Initial program 99.6%

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

      1. associate-+l- 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 79.6%

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

   if -1.4e122 < x < 1.8e16 or 1.34e73 < x < 6.59999999999999965e172

   1. Initial program 99.9%

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

      1. associate-+l- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 93.4%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+122} \lor \neg \left(x \leq 1.8 \cdot 10^{+16} \lor \neg \left(x \leq 1.34 \cdot 10^{+73}\right) \land x \leq 6.6 \cdot 10^{+172}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+124} \lor \neg \left(x \leq 1.8 \cdot 10^{+16} \lor \neg \left(x \leq 4.6 \cdot 10^{+67}\right) \land x \leq 3.6 \cdot 10^{+172}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-\left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.45e+124)
         (not (or (<= x 1.8e+16) (and (not (<= x 4.6e+67)) (<= x 3.6e+172)))))
   (* x (log y))
   (- (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.45e+124) || !((x <= 1.8e+16) || (!(x <= 4.6e+67) && (x <= 3.6e+172)))) {
		tmp = x * log(y);
	} else {
		tmp = -(y + z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-1.45d+124)) .or. (.not. (x <= 1.8d+16) .or. (.not. (x <= 4.6d+67)) .and. (x <= 3.6d+172))) then
        tmp = x * log(y)
    else
        tmp = -(y + z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.45e+124) || !((x <= 1.8e+16) || (!(x <= 4.6e+67) && (x <= 3.6e+172)))) {
		tmp = x * Math.log(y);
	} else {
		tmp = -(y + z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.45e+124) or not ((x <= 1.8e+16) or (not (x <= 4.6e+67) and (x <= 3.6e+172))):
		tmp = x * math.log(y)
	else:
		tmp = -(y + z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.45e+124) || !((x <= 1.8e+16) || (!(x <= 4.6e+67) && (x <= 3.6e+172))))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(-Float64(y + z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.45e+124) || ~(((x <= 1.8e+16) || (~((x <= 4.6e+67)) && (x <= 3.6e+172)))))
		tmp = x * log(y);
	else
		tmp = -(y + z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.45e+124], N[Not[Or[LessEqual[x, 1.8e+16], And[N[Not[LessEqual[x, 4.6e+67]], $MachinePrecision], LessEqual[x, 3.6e+172]]]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], (-N[(y + z), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < -1.45000000000000011e124 or 1.8e16 < x < 4.5999999999999997e67 or 3.59999999999999975e172 < x

   1. Initial program 99.6%

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

      1. associate-+l- 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 79.6%

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

   if -1.45000000000000011e124 < x < 1.8e16 or 4.5999999999999997e67 < x < 3.59999999999999975e172

   1. Initial program 99.9%

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

      1. associate-+l- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 84.9%

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

   6. Taylor expanded in x around 0 78.7%

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

      1. neg-mul-1 78.7%

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

      2. neg-sub0 78.7%

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

      3. metadata-eval 78.7%

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

      4. +-commutative 78.7%

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

      5. associate--r+ 78.7%

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

      6. metadata-eval 78.7%

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

      7. neg-sub0 78.7%

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

   8. Simplified 78.7%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+124} \lor \neg \left(x \leq 1.8 \cdot 10^{+16} \lor \neg \left(x \leq 4.6 \cdot 10^{+67}\right) \land x \leq 3.6 \cdot 10^{+172}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-\left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+27} \lor \neg \left(x \leq 1.8 \cdot 10^{+16}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.35e+27) (not (<= x 1.8e+16)))
   (- (* x (log y)) y)
   (- (log t) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.35e+27) || !(x <= 1.8e+16)) {
		tmp = (x * log(y)) - y;
	} else {
		tmp = log(t) - (y + z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-1.35d+27)) .or. (.not. (x <= 1.8d+16))) then
        tmp = (x * log(y)) - y
    else
        tmp = log(t) - (y + z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.35e+27) || !(x <= 1.8e+16)) {
		tmp = (x * Math.log(y)) - y;
	} else {
		tmp = Math.log(t) - (y + z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.35e+27) or not (x <= 1.8e+16):
		tmp = (x * math.log(y)) - y
	else:
		tmp = math.log(t) - (y + z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.35e+27) || !(x <= 1.8e+16))
		tmp = Float64(Float64(x * log(y)) - y);
	else
		tmp = Float64(log(t) - Float64(y + z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.35e+27) || ~((x <= 1.8e+16)))
		tmp = (x * log(y)) - y;
	else
		tmp = log(t) - (y + z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.35e+27], N[Not[LessEqual[x, 1.8e+16]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3499999999999999e27 or 1.8e16 < x

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around inf 99.7%

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

   6. Taylor expanded in z around 0 81.8%

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

   if -1.3499999999999999e27 < x < 1.8e16

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 99.3%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+27} \lor \neg \left(x \leq 1.8 \cdot 10^{+16}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -2 \cdot 10^{+26}:\\ \;\;\;\;t\_1 - y\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{+15}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -2e+26)
     (- t_1 y)
     (if (<= x 2.15e+15) (- (log t) (+ y z)) (- t_1 z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -2e+26) {
		tmp = t_1 - y;
	} else if (x <= 2.15e+15) {
		tmp = log(t) - (y + z);
	} else {
		tmp = t_1 - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if (x <= (-2d+26)) then
        tmp = t_1 - y
    else if (x <= 2.15d+15) then
        tmp = log(t) - (y + z)
    else
        tmp = t_1 - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double tmp;
	if (x <= -2e+26) {
		tmp = t_1 - y;
	} else if (x <= 2.15e+15) {
		tmp = Math.log(t) - (y + z);
	} else {
		tmp = t_1 - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -2e+26:
		tmp = t_1 - y
	elif x <= 2.15e+15:
		tmp = math.log(t) - (y + z)
	else:
		tmp = t_1 - z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -2e+26)
		tmp = Float64(t_1 - y);
	elseif (x <= 2.15e+15)
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = Float64(t_1 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (x <= -2e+26)
		tmp = t_1 - y;
	elseif (x <= 2.15e+15)
		tmp = log(t) - (y + z);
	else
		tmp = t_1 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2e+26], N[(t$95$1 - y), $MachinePrecision], If[LessEqual[x, 2.15e+15], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.0000000000000001e26

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around inf 99.7%

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

   6. Taylor expanded in z around 0 87.6%

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

   if -2.0000000000000001e26 < x < 2.15e15

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 99.3%

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

   if 2.15e15 < x

   1. Initial program 99.6%

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

      1. associate-+l- 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in z around inf 99.6%

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

   6. Taylor expanded in y around 0 91.4%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+26}:\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{+15}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - z\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -5.2 \cdot 10^{+25}:\\ \;\;\;\;t\_1 - y\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{+15}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \mathbf{else}:\\ \;\;\;\;t\_1 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -5.2e+25)
     (- t_1 y)
     (if (<= x 2.15e+15) (- (- (log t) z) y) (- t_1 z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -5.2e+25) {
		tmp = t_1 - y;
	} else if (x <= 2.15e+15) {
		tmp = (log(t) - z) - y;
	} else {
		tmp = t_1 - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if (x <= (-5.2d+25)) then
        tmp = t_1 - y
    else if (x <= 2.15d+15) then
        tmp = (log(t) - z) - y
    else
        tmp = t_1 - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double tmp;
	if (x <= -5.2e+25) {
		tmp = t_1 - y;
	} else if (x <= 2.15e+15) {
		tmp = (Math.log(t) - z) - y;
	} else {
		tmp = t_1 - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -5.2e+25:
		tmp = t_1 - y
	elif x <= 2.15e+15:
		tmp = (math.log(t) - z) - y
	else:
		tmp = t_1 - z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -5.2e+25)
		tmp = Float64(t_1 - y);
	elseif (x <= 2.15e+15)
		tmp = Float64(Float64(log(t) - z) - y);
	else
		tmp = Float64(t_1 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (x <= -5.2e+25)
		tmp = t_1 - y;
	elseif (x <= 2.15e+15)
		tmp = (log(t) - z) - y;
	else
		tmp = t_1 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.2e+25], N[(t$95$1 - y), $MachinePrecision], If[LessEqual[x, 2.15e+15], N[(N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision] - y), $MachinePrecision], N[(t$95$1 - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.1999999999999997e25

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around inf 99.7%

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

   6. Taylor expanded in z around 0 87.6%

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

   if -5.1999999999999997e25 < x < 2.15e15

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

   3. Simplified 100.0%

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

      1. add-cube-cbrt 99.2%

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

      2. pow2 99.2%

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

   6. Applied egg-rr 99.2%

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

   7. Taylor expanded in x around 0 99.3%

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

      1. neg-mul-1 99.3%

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

   9. Simplified 99.3%

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

   if 2.15e15 < x

   1. Initial program 99.6%

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

      1. associate-+l- 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in z around inf 99.6%

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

   6. Taylor expanded in y around 0 91.4%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+25}:\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{+15}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - z\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.4% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 950000000 \lor \neg \left(y \leq 1.85 \cdot 10^{+43}\right) \land y \leq 9.5 \cdot 10^{+115}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y 950000000.0) (and (not (<= y 1.85e+43)) (<= y 9.5e+115)))
   (- z)
   (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= 950000000.0) || (!(y <= 1.85e+43) && (y <= 9.5e+115))) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= 950000000.0d0) .or. (.not. (y <= 1.85d+43)) .and. (y <= 9.5d+115)) then
        tmp = -z
    else
        tmp = -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= 950000000.0) || (!(y <= 1.85e+43) && (y <= 9.5e+115))) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= 950000000.0) or (not (y <= 1.85e+43) and (y <= 9.5e+115)):
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= 950000000.0) || (!(y <= 1.85e+43) && (y <= 9.5e+115)))
		tmp = Float64(-z);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= 950000000.0) || (~((y <= 1.85e+43)) && (y <= 9.5e+115)))
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, 950000000.0], And[N[Not[LessEqual[y, 1.85e+43]], $MachinePrecision], LessEqual[y, 9.5e+115]]], (-z), (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 9.5e8 or 1.85e43 < y < 9.4999999999999997e115

   1. Initial program 99.8%

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

      1. associate-+l- 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 46.8%

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

      1. neg-mul-1 46.8%

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

   7. Simplified 46.8%

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

   if 9.5e8 < y < 1.85e43 or 9.4999999999999997e115 < y

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around inf 69.4%

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

      1. neg-mul-1 69.4%

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

   7. Simplified 69.4%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 950000000 \lor \neg \left(y \leq 1.85 \cdot 10^{+43}\right) \land y \leq 9.5 \cdot 10^{+115}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.7% accurate, 52.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-+l- 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in z around inf 88.9%

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

6. Taylor expanded in x around 0 62.4%

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

   1. neg-mul-1 62.4%

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

   2. neg-sub0 62.4%

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

   3. metadata-eval 62.4%

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

   4. +-commutative 62.4%

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

   5. associate--r+ 62.4%

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

   6. metadata-eval 62.4%

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

   7. neg-sub0 62.4%

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

8. Simplified 62.4%

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

9. Final simplification 62.4%

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

Alternative 14: 30.0% accurate, 104.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-+l- 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in y around inf 29.4%

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

   1. neg-mul-1 29.4%

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

7. Simplified 29.4%

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

8. Final simplification 29.4%

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

Reproduce

herbie shell --seed 2024062 
(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)))