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

Percentage Accurate: 99.9% → 99.9%

Time: 8.1s

Alternatives: 7

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. *-rgt-identity N/A

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

   4. *-inverses N/A

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

   5. fp-cancel-sign-sub N/A

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

   6. mul-1-neg N/A

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

   7. associate-/l* N/A

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

   8. associate-*l/ N/A

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

   9. associate-*r/ N/A

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

   10. *-commutative N/A

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

   11. remove-double-neg N/A

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

   12. fp-cancel-sign-sub N/A

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

   13. mul-1-neg N/A

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

   14. *-commutative N/A

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

5. Applied rewrites 99.9%

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

Alternative 2: 68.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - y\\ \mathbf{if}\;t\_1 \leq -10000:\\ \;\;\;\;\log t - y\\ \mathbf{elif}\;t\_1 \leq 10^{+23}:\\ \;\;\;\;\left(-z\right) + \log t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) y)))
   (if (<= t_1 -10000.0)
     (- (log t) y)
     (if (<= t_1 1e+23) (+ (- z) (log t)) (* (log y) x)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - y;
	double tmp;
	if (t_1 <= -10000.0) {
		tmp = log(t) - y;
	} else if (t_1 <= 1e+23) {
		tmp = -z + log(t);
	} else {
		tmp = log(y) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * log(y)) - y
    if (t_1 <= (-10000.0d0)) then
        tmp = log(t) - y
    else if (t_1 <= 1d+23) then
        tmp = -z + log(t)
    else
        tmp = log(y) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * Math.log(y)) - y;
	double tmp;
	if (t_1 <= -10000.0) {
		tmp = Math.log(t) - y;
	} else if (t_1 <= 1e+23) {
		tmp = -z + Math.log(t);
	} else {
		tmp = Math.log(y) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - y
	tmp = 0
	if t_1 <= -10000.0:
		tmp = math.log(t) - y
	elif t_1 <= 1e+23:
		tmp = -z + math.log(t)
	else:
		tmp = math.log(y) * x
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - y)
	tmp = 0.0
	if (t_1 <= -10000.0)
		tmp = Float64(log(t) - y);
	elseif (t_1 <= 1e+23)
		tmp = Float64(Float64(-z) + log(t));
	else
		tmp = Float64(log(y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * log(y)) - y;
	tmp = 0.0;
	if (t_1 <= -10000.0)
		tmp = log(t) - y;
	elseif (t_1 <= 1e+23)
		tmp = -z + log(t);
	else
		tmp = log(y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t$95$1, -10000.0], N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision], If[LessEqual[t$95$1, 1e+23], N[((-z) + N[Log[t], $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -1e4

   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 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 73.4

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

   5. Applied rewrites 73.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 51.4%

      \[\leadsto \log t - y \]

   if -1e4 < (-.f64 (*.f64 x (log.f64 y)) y) < 9.9999999999999992e22

   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--.f64 N/A

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

      2. flip-- N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-1 \cdot z} + \log t \]
   8. 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 98.5

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

   9. Applied rewrites 98.5%

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

   if 9.9999999999999992e22 < (-.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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 91.7%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left({\log y}^{2} \cdot x, \color{blue}{\frac{1}{\log y}}, \frac{y}{\mathsf{fma}\left(\log y, x, y\right)} \cdot \left(-y\right) - z\right) + \log t \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left({\log y}^{2} \cdot x, \color{blue}{\frac{1}{\log y}}, \frac{y}{\mathsf{fma}\left(\log y, x, y\right)} \cdot \left(-y\right) - z\right) + \log t \]

      2. lower-log.f64 99.3

         \[\leadsto \mathsf{fma}\left({\log y}^{2} \cdot x, \frac{1}{\color{blue}{\log y}}, \frac{y}{\mathsf{fma}\left(\log y, x, y\right)} \cdot \left(-y\right) - z\right) + \log t \]

   9. Applied rewrites 99.3%

      \[\leadsto \mathsf{fma}\left({\log y}^{2} \cdot x, \color{blue}{\frac{1}{\log y}}, \frac{y}{\mathsf{fma}\left(\log y, x, y\right)} \cdot \left(-y\right) - z\right) + \log t \]

   10. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-log.f64 79.0

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

   12. Applied rewrites 79.0%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -120000000 \lor \neg \left(z \leq 1.4 \cdot 10^{-7}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \left(\frac{-y}{z} - 1\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -120000000.0) (not (<= z 1.4e-7)))
   (fma (log y) x (* (- (/ (- y) z) 1.0) z))
   (- (fma (log y) x (log t)) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -120000000.0) || !(z <= 1.4e-7)) {
		tmp = fma(log(y), x, (((-y / z) - 1.0) * z));
	} else {
		tmp = fma(log(y), x, log(t)) - y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -120000000.0) || !(z <= 1.4e-7))
		tmp = fma(log(y), x, Float64(Float64(Float64(Float64(-y) / z) - 1.0) * z));
	else
		tmp = Float64(fma(log(y), x, log(t)) - y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -120000000.0], N[Not[LessEqual[z, 1.4e-7]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x + N[(N[(N[((-y) / z), $MachinePrecision] - 1.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * x + N[Log[t], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2e8 or 1.4000000000000001e-7 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-rgt-identity N/A

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

      4. *-inverses N/A

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

      5. fp-cancel-sign-sub N/A

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

      6. mul-1-neg N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. remove-double-neg N/A

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

      12. fp-cancel-sign-sub N/A

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

      13. mul-1-neg N/A

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

      14. *-commutative N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 84.8%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 99.9%

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

   12. Taylor expanded in y around inf

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

   14. Applied rewrites 99.4%

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

   if -1.2e8 < z < 1.4000000000000001e-7

   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 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 98.9

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

   5. Applied rewrites 98.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -120000000 \lor \neg \left(z \leq 1.4 \cdot 10^{-7}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \left(\frac{-y}{z} - 1\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t\right) - y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+20} \lor \neg \left(x \leq 2 \cdot 10^{+21}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \left(\frac{-y}{z} - 1\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -4e+20) (not (<= x 2e+21)))
   (fma (log y) x (* (- (/ (- y) z) 1.0) z))
   (- (- (log t) y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4e+20) || !(x <= 2e+21)) {
		tmp = fma(log(y), x, (((-y / z) - 1.0) * z));
	} else {
		tmp = (log(t) - y) - z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -4e+20) || !(x <= 2e+21))
		tmp = fma(log(y), x, Float64(Float64(Float64(Float64(-y) / z) - 1.0) * z));
	else
		tmp = Float64(Float64(log(t) - y) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -4e+20], N[Not[LessEqual[x, 2e+21]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x + N[(N[(N[((-y) / z), $MachinePrecision] - 1.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4e20 or 2e21 < 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 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-rgt-identity N/A

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

      4. *-inverses N/A

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

      5. fp-cancel-sign-sub N/A

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

      6. mul-1-neg N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. remove-double-neg N/A

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

      12. fp-cancel-sign-sub N/A

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

      13. mul-1-neg N/A

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

      14. *-commutative N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 81.4%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 90.4%

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

   12. Taylor expanded in y around inf

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

   14. Applied rewrites 90.4%

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

   if -4e20 < x < 2e21

   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. associate--r+ N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. lower--.f64 N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 98.3

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

   5. Applied rewrites 98.3%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+20} \lor \neg \left(x \leq 2 \cdot 10^{+21}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \left(\frac{-y}{z} - 1\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+107} \lor \neg \left(x \leq 10^{+126}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -8e+107) (not (<= x 1e+126)))
   (* (log y) x)
   (- (- (log t) y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -8e+107) || !(x <= 1e+126)) {
		tmp = log(y) * x;
	} else {
		tmp = (log(t) - y) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-8d+107)) .or. (.not. (x <= 1d+126))) then
        tmp = log(y) * x
    else
        tmp = (log(t) - y) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -8e+107) || !(x <= 1e+126)) {
		tmp = Math.log(y) * x;
	} else {
		tmp = (Math.log(t) - y) - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -8e+107) or not (x <= 1e+126):
		tmp = math.log(y) * x
	else:
		tmp = (math.log(t) - y) - z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -8e+107) || !(x <= 1e+126))
		tmp = Float64(log(y) * x);
	else
		tmp = Float64(Float64(log(t) - y) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -8e+107) || ~((x <= 1e+126)))
		tmp = log(y) * x;
	else
		tmp = (log(t) - y) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -8e+107], N[Not[LessEqual[x, 1e+126]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.9999999999999998e107 or 9.99999999999999925e125 < x

   1. Initial program 99.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 81.6%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left({\log y}^{2} \cdot x, \color{blue}{\frac{1}{\log y}}, \frac{y}{\mathsf{fma}\left(\log y, x, y\right)} \cdot \left(-y\right) - z\right) + \log t \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left({\log y}^{2} \cdot x, \color{blue}{\frac{1}{\log y}}, \frac{y}{\mathsf{fma}\left(\log y, x, y\right)} \cdot \left(-y\right) - z\right) + \log t \]

      2. lower-log.f64 98.5

         \[\leadsto \mathsf{fma}\left({\log y}^{2} \cdot x, \frac{1}{\color{blue}{\log y}}, \frac{y}{\mathsf{fma}\left(\log y, x, y\right)} \cdot \left(-y\right) - z\right) + \log t \]

   9. Applied rewrites 98.5%

      \[\leadsto \mathsf{fma}\left({\log y}^{2} \cdot x, \color{blue}{\frac{1}{\log y}}, \frac{y}{\mathsf{fma}\left(\log y, x, y\right)} \cdot \left(-y\right) - z\right) + \log t \]

   10. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-log.f64 78.1

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

   12. Applied rewrites 78.1%

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

   if -7.9999999999999998e107 < x < 9.99999999999999925e125

   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. associate--r+ N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. lower--.f64 N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 92.1

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

   5. Applied rewrites 92.1%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+107} \lor \neg \left(x \leq 10^{+126}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+27} \lor \neg \left(z \leq 2.6 \cdot 10^{+142}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\log t - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.15e+27) (not (<= z 2.6e+142))) (- z) (- (log t) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.15e+27) || !(z <= 2.6e+142)) {
		tmp = -z;
	} else {
		tmp = log(t) - y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-1.15d+27)) .or. (.not. (z <= 2.6d+142))) then
        tmp = -z
    else
        tmp = log(t) - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.15e+27) || !(z <= 2.6e+142)) {
		tmp = -z;
	} else {
		tmp = Math.log(t) - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.15e+27) or not (z <= 2.6e+142):
		tmp = -z
	else:
		tmp = math.log(t) - y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.15e+27) || !(z <= 2.6e+142))
		tmp = Float64(-z);
	else
		tmp = Float64(log(t) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.15e+27) || ~((z <= 2.6e+142)))
		tmp = -z;
	else
		tmp = log(t) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.15e+27], N[Not[LessEqual[z, 2.6e+142]], $MachinePrecision]], (-z), N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.15e27 or 2.60000000000000021e142 < 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 76.1

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

   5. Applied rewrites 76.1%

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

   if -1.15e27 < z < 2.60000000000000021e142

   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 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 93.5

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 58.3%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+27} \lor \neg \left(z \leq 2.6 \cdot 10^{+142}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\log t - y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 28.8% accurate, 71.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 35.6

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

5. Applied rewrites 35.6%

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

Reproduce

herbie shell --seed 2024338 
(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)))