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

Percentage Accurate: 99.9% → 99.9%

Time: 9.5s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 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. sub-neg N/A

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

   4. sub-neg N/A

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

   5. remove-double-neg N/A

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

   6. mul-1-neg N/A

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

   7. *-commutative N/A

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

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

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

   9. metadata-eval N/A

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

   10. cancel-sign-sub N/A

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

   11. mul-1-neg N/A

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

   12. *-inverses N/A

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

   13. associate-/l* N/A

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

   14. associate-*l/ N/A

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

   15. associate-*r/ N/A

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

   16. sub-neg N/A

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

5. Applied rewrites 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 73.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - y\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+253}:\\ \;\;\;\;{\left(\frac{-1}{y}\right)}^{-1}\\ \mathbf{elif}\;t\_1 \leq -5000000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{-y}{z}, z, -z\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+20}:\\ \;\;\;\;\left(-z\right) + \log t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) y)))
   (if (<= t_1 -2e+253)
     (pow (/ -1.0 y) -1.0)
     (if (<= t_1 -5000000000000.0)
       (fma (/ (- y) z) z (- z))
       (if (<= t_1 2e+20) (+ (- 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 <= -2e+253) {
		tmp = pow((-1.0 / y), -1.0);
	} else if (t_1 <= -5000000000000.0) {
		tmp = fma((-y / z), z, -z);
	} else if (t_1 <= 2e+20) {
		tmp = -z + log(t);
	} else {
		tmp = 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 <= -2e+253)
		tmp = Float64(-1.0 / y) ^ -1.0;
	elseif (t_1 <= -5000000000000.0)
		tmp = fma(Float64(Float64(-y) / z), z, Float64(-z));
	elseif (t_1 <= 2e+20)
		tmp = Float64(Float64(-z) + log(t));
	else
		tmp = Float64(log(y) * x);
	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, -2e+253], N[Power[N[(-1.0 / y), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[t$95$1, -5000000000000.0], N[(N[((-y) / z), $MachinePrecision] * z + (-z)), $MachinePrecision], If[LessEqual[t$95$1, 2e+20], N[((-z) + N[Log[t], $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -1.9999999999999999e253

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 61.8

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

   7. Applied rewrites 61.8%

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

   if -1.9999999999999999e253 < (-.f64 (*.f64 x (log.f64 y)) y) < -5e12

   1. Initial program 99.9%

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

   3. Taylor expanded in 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. sub-neg N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

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

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. *-inverses N/A

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

      13. associate-/l* N/A

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

      14. associate-*l/ N/A

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

      15. associate-*r/ N/A

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

      16. sub-neg N/A

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

   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 z around -inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. neg-mul-1 N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. distribute-lft-neg-out N/A

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

      9. remove-double-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 80.4%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 58.7%

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

   if -5e12 < (-.f64 (*.f64 x (log.f64 y)) y) < 2e20

   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

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

      2. sub-neg N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. div-inv N/A

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

      6. difference-of-squares N/A

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

      7. lift--.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 97.2

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

   7. Applied rewrites 97.2%

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

   if 2e20 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.6%

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

   3. 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. sub-neg N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

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

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. *-inverses N/A

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

      13. associate-/l* N/A

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

      14. associate-*l/ N/A

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

      15. associate-*r/ N/A

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

      16. sub-neg N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} \]
   7. 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 86.3

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

   8. Applied rewrites 86.3%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log y - y \leq -2 \cdot 10^{+253}:\\ \;\;\;\;{\left(\frac{-1}{y}\right)}^{-1}\\ \mathbf{elif}\;x \cdot \log y - y \leq -5000000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{-y}{z}, z, -z\right)\\ \mathbf{elif}\;x \cdot \log y - y \leq 2 \cdot 10^{+20}:\\ \;\;\;\;\left(-z\right) + \log t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+98} \lor \neg \left(z \leq 2.45 \cdot 10^{+20}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{\log y \cdot x}{z}, z, -z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.3e+98) (not (<= z 2.45e+20)))
   (fma (/ (* (log y) x) z) z (- z))
   (fma (log y) x (- (log t) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.3e+98) || !(z <= 2.45e+20)) {
		tmp = fma(((log(y) * x) / z), z, -z);
	} else {
		tmp = fma(log(y), x, (log(t) - y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.3e+98) || !(z <= 2.45e+20))
		tmp = fma(Float64(Float64(log(y) * x) / z), z, Float64(-z));
	else
		tmp = fma(log(y), x, Float64(log(t) - y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.3e+98], N[Not[LessEqual[z, 2.45e+20]], $MachinePrecision]], N[(N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision] * z + (-z)), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x + N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3e98 or 2.45e20 < z

   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

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

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

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

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. *-inverses N/A

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

      13. associate-/l* N/A

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

      14. associate-*l/ N/A

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

      15. associate-*r/ N/A

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

      16. sub-neg N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around -inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. neg-mul-1 N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. distribute-lft-neg-out N/A

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

      9. remove-double-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 86.1%

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

   if -1.3e98 < z < 2.45e20

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 98.8

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

   5. Applied rewrites 98.8%

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

   7. Applied rewrites 98.8%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+98} \lor \neg \left(z \leq 2.45 \cdot 10^{+20}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{\log y \cdot x}{z}, z, -z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+98} \lor \neg \left(z \leq 2.45 \cdot 10^{+20}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{\log y \cdot x}{z}, z, -z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.3e+98) (not (<= z 2.45e+20)))
   (fma (/ (* (log y) x) z) z (- z))
   (- (fma (log y) x (log t)) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.3e+98) || !(z <= 2.45e+20)) {
		tmp = fma(((log(y) * x) / z), z, -z);
	} else {
		tmp = fma(log(y), x, log(t)) - y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.3e+98) || !(z <= 2.45e+20))
		tmp = fma(Float64(Float64(log(y) * x) / z), z, Float64(-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, -1.3e+98], N[Not[LessEqual[z, 2.45e+20]], $MachinePrecision]], N[(N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision] * z + (-z)), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * x + N[Log[t], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3e98 or 2.45e20 < z

   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

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

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

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

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. *-inverses N/A

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

      13. associate-/l* N/A

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

      14. associate-*l/ N/A

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

      15. associate-*r/ N/A

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

      16. sub-neg N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around -inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. neg-mul-1 N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. distribute-lft-neg-out N/A

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

      9. remove-double-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 86.1%

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

   if -1.3e98 < z < 2.45e20

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 98.8

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

   5. Applied rewrites 98.8%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+98} \lor \neg \left(z \leq 2.45 \cdot 10^{+20}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{\log y \cdot x}{z}, z, -z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t\right) - y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ t_2 := \frac{-y}{z}\\ \mathbf{if}\;z \leq -1.05 \cdot 10^{+87}:\\ \;\;\;\;z \cdot \left(t\_2 + -1\right)\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{-51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\log t - y\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-22}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, z, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)) (t_2 (/ (- y) z)))
   (if (<= z -1.05e+87)
     (* z (+ t_2 -1.0))
     (if (<= z -1.4e-51)
       t_1
       (if (<= z 1.55e-162)
         (- (log t) y)
         (if (<= z 7.8e-22) t_1 (fma t_2 z (- z))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double t_2 = -y / z;
	double tmp;
	if (z <= -1.05e+87) {
		tmp = z * (t_2 + -1.0);
	} else if (z <= -1.4e-51) {
		tmp = t_1;
	} else if (z <= 1.55e-162) {
		tmp = log(t) - y;
	} else if (z <= 7.8e-22) {
		tmp = t_1;
	} else {
		tmp = fma(t_2, z, -z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	t_2 = Float64(Float64(-y) / z)
	tmp = 0.0
	if (z <= -1.05e+87)
		tmp = Float64(z * Float64(t_2 + -1.0));
	elseif (z <= -1.4e-51)
		tmp = t_1;
	elseif (z <= 1.55e-162)
		tmp = Float64(log(t) - y);
	elseif (z <= 7.8e-22)
		tmp = t_1;
	else
		tmp = fma(t_2, z, Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[((-y) / z), $MachinePrecision]}, If[LessEqual[z, -1.05e+87], N[(z * N[(t$95$2 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.4e-51], t$95$1, If[LessEqual[z, 1.55e-162], N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision], If[LessEqual[z, 7.8e-22], t$95$1, N[(t$95$2 * z + (-z)), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.05e87

   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. sub-neg N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

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

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. *-inverses N/A

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

      13. associate-/l* N/A

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

      14. associate-*l/ N/A

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

      15. associate-*r/ N/A

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

      16. sub-neg N/A

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

   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 z around -inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. neg-mul-1 N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. distribute-lft-neg-out N/A

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

      9. remove-double-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 84.6%

       \[\leadsto \mathsf{fma}\left(\frac{-y}{z}, z, -z\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 84.6%

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

   if -1.05e87 < z < -1.4e-51 or 1.5499999999999999e-162 < z < 7.79999999999999996e-22

   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. sub-neg N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

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

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. *-inverses N/A

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

      13. associate-/l* N/A

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

      14. associate-*l/ N/A

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

      15. associate-*r/ N/A

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

      16. sub-neg N/A

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

   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 x around inf

      \[\leadsto \color{blue}{x \cdot \log y} \]
   7. 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 65.5

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

   8. Applied rewrites 65.5%

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

   if -1.4e-51 < z < 1.5499999999999999e-162

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 61.6%

      \[\leadsto \log t - y \]

   if 7.79999999999999996e-22 < 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. sub-neg N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

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

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. *-inverses N/A

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

      13. associate-/l* N/A

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

      14. associate-*l/ N/A

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

      15. associate-*r/ N/A

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

      16. sub-neg N/A

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

   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 z around -inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. neg-mul-1 N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. distribute-lft-neg-out N/A

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

      9. remove-double-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 79.7%

       \[\leadsto \mathsf{fma}\left(\frac{-y}{z}, z, -z\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+87}:\\ \;\;\;\;z \cdot \left(\frac{-y}{z} + -1\right)\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{-51}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\log t - y\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-22}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-y}{z}, z, -z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+136} \lor \neg \left(x \leq 8.5 \cdot 10^{+136}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -6.8e+136) (not (<= x 8.5e+136)))
   (* (log y) x)
   (- (- (log t) y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -6.8e+136) || !(x <= 8.5e+136)) {
		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 <= (-6.8d+136)) .or. (.not. (x <= 8.5d+136))) 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 <= -6.8e+136) || !(x <= 8.5e+136)) {
		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 <= -6.8e+136) or not (x <= 8.5e+136):
		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 <= -6.8e+136) || !(x <= 8.5e+136))
		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 <= -6.8e+136) || ~((x <= 8.5e+136)))
		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, -6.8e+136], N[Not[LessEqual[x, 8.5e+136]], $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 < -6.79999999999999993e136 or 8.49999999999999966e136 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 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. sub-neg N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

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

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. *-inverses N/A

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

      13. associate-/l* N/A

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

      14. associate-*l/ N/A

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

      15. associate-*r/ N/A

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

      16. sub-neg N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} \]
   7. 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

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

   8. Applied rewrites 78.1%

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

   if -6.79999999999999993e136 < x < 8.49999999999999966e136

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 88.2

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

   5. Applied rewrites 88.2%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+136} \lor \neg \left(x \leq 8.5 \cdot 10^{+136}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+62} \lor \neg \left(z \leq 420\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-y}{z}, z, -z\right)\\ \mathbf{else}:\\ \;\;\;\;\log t - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.3e+62) (not (<= z 420.0)))
   (fma (/ (- y) z) z (- z))
   (- (log t) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.3e+62) || !(z <= 420.0)) {
		tmp = fma((-y / z), z, -z);
	} else {
		tmp = log(t) - y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.3e+62) || !(z <= 420.0))
		tmp = fma(Float64(Float64(-y) / z), z, Float64(-z));
	else
		tmp = Float64(log(t) - y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.3e+62], N[Not[LessEqual[z, 420.0]], $MachinePrecision]], N[(N[((-y) / z), $MachinePrecision] * z + (-z)), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.29999999999999992e62 or 420 < z

   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

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

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

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

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. *-inverses N/A

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

      13. associate-/l* N/A

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

      14. associate-*l/ N/A

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

      15. associate-*r/ N/A

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

      16. sub-neg N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around -inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. neg-mul-1 N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. distribute-lft-neg-out N/A

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

      9. remove-double-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 80.2%

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

   if -1.29999999999999992e62 < z < 420

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.5%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+62} \lor \neg \left(z \leq 420\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-y}{z}, z, -z\right)\\ \mathbf{else}:\\ \;\;\;\;\log t - y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.2% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-25} \lor \neg \left(z \leq 5.4 \cdot 10^{-14}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-y}{z}, z, -z\right)\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.55e-25) (not (<= z 5.4e-14)))
   (fma (/ (- y) z) z (- z))
   (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.55e-25) || !(z <= 5.4e-14)) {
		tmp = fma((-y / z), z, -z);
	} else {
		tmp = -y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.55e-25) || !(z <= 5.4e-14))
		tmp = fma(Float64(Float64(-y) / z), z, Float64(-z));
	else
		tmp = Float64(-y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.55e-25], N[Not[LessEqual[z, 5.4e-14]], $MachinePrecision]], N[(N[((-y) / z), $MachinePrecision] * z + (-z)), $MachinePrecision], (-y)]

Derivation

1. Split input into 2 regimes

2. if z < -1.54999999999999997e-25 or 5.3999999999999997e-14 < 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. sub-neg N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

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

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. *-inverses N/A

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

      13. associate-/l* N/A

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

      14. associate-*l/ N/A

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

      15. associate-*r/ N/A

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

      16. sub-neg N/A

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

   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 z around -inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. neg-mul-1 N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. distribute-lft-neg-out N/A

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

      9. remove-double-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 74.1%

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

   if -1.54999999999999997e-25 < z < 5.3999999999999997e-14

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 34.1

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

   7. Applied rewrites 34.1%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-25} \lor \neg \left(z \leq 5.4 \cdot 10^{-14}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-y}{z}, z, -z\right)\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-25} \lor \neg \left(z \leq 5.4 \cdot 10^{-14}\right):\\ \;\;\;\;z \cdot \left(\frac{-y}{z} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.55e-25) (not (<= z 5.4e-14)))
   (* z (+ (/ (- y) z) -1.0))
   (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.55e-25) || !(z <= 5.4e-14)) {
		tmp = z * ((-y / z) + -1.0);
	} 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 ((z <= (-1.55d-25)) .or. (.not. (z <= 5.4d-14))) then
        tmp = z * ((-y / z) + (-1.0d0))
    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 ((z <= -1.55e-25) || !(z <= 5.4e-14)) {
		tmp = z * ((-y / z) + -1.0);
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.55e-25) or not (z <= 5.4e-14):
		tmp = z * ((-y / z) + -1.0)
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.55e-25) || !(z <= 5.4e-14))
		tmp = Float64(z * Float64(Float64(Float64(-y) / z) + -1.0));
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.55e-25) || ~((z <= 5.4e-14)))
		tmp = z * ((-y / z) + -1.0);
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.55e-25], N[Not[LessEqual[z, 5.4e-14]], $MachinePrecision]], N[(z * N[(N[((-y) / z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], (-y)]

Derivation

1. Split input into 2 regimes

2. if z < -1.54999999999999997e-25 or 5.3999999999999997e-14 < 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. sub-neg N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

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

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. *-inverses N/A

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

      13. associate-/l* N/A

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

      14. associate-*l/ N/A

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

      15. associate-*r/ N/A

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

      16. sub-neg N/A

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

   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 z around -inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. neg-mul-1 N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. distribute-lft-neg-out N/A

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

      9. remove-double-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 74.1%

       \[\leadsto \mathsf{fma}\left(\frac{-y}{z}, z, -z\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 74.1%

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

   if -1.54999999999999997e-25 < z < 5.3999999999999997e-14

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 34.1

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

   7. Applied rewrites 34.1%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-25} \lor \neg \left(z \leq 5.4 \cdot 10^{-14}\right):\\ \;\;\;\;z \cdot \left(\frac{-y}{z} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.9% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+98} \lor \neg \left(z \leq 2 \cdot 10^{+20}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.3e+98) (not (<= z 2e+20))) (- z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.3e+98) || !(z <= 2e+20)) {
		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 ((z <= (-1.3d+98)) .or. (.not. (z <= 2d+20))) 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 ((z <= -1.3e+98) || !(z <= 2e+20)) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.3e+98) or not (z <= 2e+20):
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.3e+98) || !(z <= 2e+20))
		tmp = Float64(-z);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.3e+98) || ~((z <= 2e+20)))
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.3e+98], N[Not[LessEqual[z, 2e+20]], $MachinePrecision]], (-z), (-y)]

Derivation

1. Split input into 2 regimes

2. if z < -1.3e98 or 2e20 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 70.0

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

   5. Applied rewrites 70.0%

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

   if -1.3e98 < z < 2e20

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 34.6

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

   7. Applied rewrites 34.6%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+98} \lor \neg \left(z \leq 2 \cdot 10^{+20}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 30.1% 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 31.0

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

5. Applied rewrites 31.0%

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

6. Final simplification 31.0%

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

Alternative 12: 2.2% accurate, 215.0× 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 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 31.0

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

5. Applied rewrites 31.0%

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

7. Applied rewrites 12.1%

   \[\leadsto \frac{0 - z \cdot z}{\color{blue}{0 + z}} \]
8. Step-by-step derivation

9. Applied rewrites 2.2%

   \[\leadsto z \]

10. Final simplification 2.2%

    \[\leadsto z \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024313 
(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)))