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

Percentage Accurate: 99.9% → 99.9%

Time: 9.1s

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. remove-double-neg N/A

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

   4. mul-1-neg N/A

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

   5. *-commutative N/A

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

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

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

   7. metadata-eval N/A

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

   8. cancel-sign-sub N/A

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

   9. mul-1-neg N/A

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

   10. *-inverses N/A

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

   11. associate-/l* N/A

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

   12. associate-*l/ N/A

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

   13. associate-*r/ N/A

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

   14. sub-neg N/A

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

   15. *-commutative N/A

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

   16. lower-fma.f64 N/A

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

5. Applied rewrites 99.9%

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

Alternative 2: 67.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ t_2 := t\_1 - y\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+28}:\\ \;\;\;\;-y\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+113}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)) (t_2 (- t_1 y)))
   (if (<= t_2 -2e+28) (- y) (if (<= t_2 5e+113) (- (log t) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double t_2 = t_1 - y;
	double tmp;
	if (t_2 <= -2e+28) {
		tmp = -y;
	} else if (t_2 <= 5e+113) {
		tmp = log(t) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = log(y) * x
    t_2 = t_1 - y
    if (t_2 <= (-2d+28)) then
        tmp = -y
    else if (t_2 <= 5d+113) then
        tmp = log(t) - z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) * x;
	double t_2 = t_1 - y;
	double tmp;
	if (t_2 <= -2e+28) {
		tmp = -y;
	} else if (t_2 <= 5e+113) {
		tmp = Math.log(t) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	t_2 = t_1 - y
	tmp = 0
	if t_2 <= -2e+28:
		tmp = -y
	elif t_2 <= 5e+113:
		tmp = math.log(t) - z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	t_2 = Float64(t_1 - y)
	tmp = 0.0
	if (t_2 <= -2e+28)
		tmp = Float64(-y);
	elseif (t_2 <= 5e+113)
		tmp = Float64(log(t) - z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	t_2 = t_1 - y;
	tmp = 0.0;
	if (t_2 <= -2e+28)
		tmp = -y;
	elseif (t_2 <= 5e+113)
		tmp = log(t) - z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - y), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+28], (-y), If[LessEqual[t$95$2, 5e+113], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -1.99999999999999992e28

   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. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. cancel-sign-sub N/A

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

      9. mul-1-neg N/A

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

      10. *-inverses N/A

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

      11. associate-/l* N/A

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

      12. associate-*l/ N/A

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

      13. associate-*r/ N/A

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

      14. sub-neg N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

   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 y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 52.4

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

   8. Applied rewrites 52.4%

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

   if -1.99999999999999992e28 < (-.f64 (*.f64 x (log.f64 y)) y) < 5e113

   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. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. cancel-sign-sub N/A

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

      9. mul-1-neg N/A

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

      10. *-inverses N/A

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

      11. associate-/l* N/A

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

      12. associate-*l/ N/A

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

      13. associate-*r/ N/A

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

      14. sub-neg N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

   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 x around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 91.4

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

   8. Applied rewrites 91.4%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 90.2%

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

   if 5e113 < (-.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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} \]
   6. 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 87.4

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

   7. Applied rewrites 87.4%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log y \cdot x - y \leq -2 \cdot 10^{+28}:\\ \;\;\;\;-y\\ \mathbf{elif}\;\log y \cdot x - y \leq 5 \cdot 10^{+113}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t - y\\ \mathbf{if}\;z \leq -4.5 \cdot 10^{+88}:\\ \;\;\;\;t\_1 - z\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (log t) y)))
   (if (<= z -4.5e+88)
     (- t_1 z)
     (if (<= z 3.3e-10) (fma (log y) x t_1) (fma (log y) x (- (log t) z))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(t) - y;
	double tmp;
	if (z <= -4.5e+88) {
		tmp = t_1 - z;
	} else if (z <= 3.3e-10) {
		tmp = fma(log(y), x, t_1);
	} else {
		tmp = fma(log(y), x, (log(t) - z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(log(t) - y)
	tmp = 0.0
	if (z <= -4.5e+88)
		tmp = Float64(t_1 - z);
	elseif (z <= 3.3e-10)
		tmp = fma(log(y), x, t_1);
	else
		tmp = fma(log(y), x, Float64(log(t) - z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[z, -4.5e+88], N[(t$95$1 - z), $MachinePrecision], If[LessEqual[z, 3.3e-10], N[(N[Log[y], $MachinePrecision] * x + t$95$1), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x + N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.5e88

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate--r+ N/A

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

      2. 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 90.2

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

   5. Applied rewrites 90.2%

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

   if -4.5e88 < z < 3.3e-10

   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 97.8

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

   5. Applied rewrites 97.8%

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

   7. Applied rewrites 97.8%

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

   if 3.3e-10 < 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. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. cancel-sign-sub N/A

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

      9. mul-1-neg N/A

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

      10. *-inverses N/A

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

      11. associate-/l* N/A

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

      12. associate-*l/ N/A

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

      13. associate-*r/ N/A

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

      14. sub-neg N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

   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 y around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 88.2

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

   8. Applied rewrites 88.2%

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

Alternative 4: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t - y\\ \mathbf{if}\;z \leq -4.5 \cdot 10^{+88}:\\ \;\;\;\;t\_1 - z\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (log t) y)))
   (if (<= z -4.5e+88)
     (- t_1 z)
     (if (<= z 3.3e-10) (fma (log y) x t_1) (- (fma (log y) x (log t)) z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(t) - y;
	double tmp;
	if (z <= -4.5e+88) {
		tmp = t_1 - z;
	} else if (z <= 3.3e-10) {
		tmp = fma(log(y), x, t_1);
	} else {
		tmp = fma(log(y), x, log(t)) - z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(log(t) - y)
	tmp = 0.0
	if (z <= -4.5e+88)
		tmp = Float64(t_1 - z);
	elseif (z <= 3.3e-10)
		tmp = fma(log(y), x, t_1);
	else
		tmp = Float64(fma(log(y), x, log(t)) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[z, -4.5e+88], N[(t$95$1 - z), $MachinePrecision], If[LessEqual[z, 3.3e-10], N[(N[Log[y], $MachinePrecision] * x + t$95$1), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * x + N[Log[t], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.5e88

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate--r+ N/A

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

      2. 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 90.2

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

   5. Applied rewrites 90.2%

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

   if -4.5e88 < z < 3.3e-10

   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 97.8

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

   5. Applied rewrites 97.8%

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

   7. Applied rewrites 97.8%

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

   if 3.3e-10 < 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 y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 88.2

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

   5. Applied rewrites 88.2%

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

Alternative 5: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\log y, x, \log t\right)\\ \mathbf{if}\;z \leq -4.5 \cdot 10^{+88}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-10}:\\ \;\;\;\;t\_1 - y\\ \mathbf{else}:\\ \;\;\;\;t\_1 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (log y) x (log t))))
   (if (<= z -4.5e+88)
     (- (- (log t) y) z)
     (if (<= z 3.3e-10) (- t_1 y) (- t_1 z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(log(y), x, log(t));
	double tmp;
	if (z <= -4.5e+88) {
		tmp = (log(t) - y) - z;
	} else if (z <= 3.3e-10) {
		tmp = t_1 - y;
	} else {
		tmp = t_1 - z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(log(y), x, log(t))
	tmp = 0.0
	if (z <= -4.5e+88)
		tmp = Float64(Float64(log(t) - y) - z);
	elseif (z <= 3.3e-10)
		tmp = Float64(t_1 - y);
	else
		tmp = Float64(t_1 - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x + N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.5e+88], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[z, 3.3e-10], N[(t$95$1 - y), $MachinePrecision], N[(t$95$1 - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.5e88

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate--r+ N/A

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

      2. 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 90.2

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

   5. Applied rewrites 90.2%

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

   if -4.5e88 < z < 3.3e-10

   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 97.8

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

   5. Applied rewrites 97.8%

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

   if 3.3e-10 < 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 y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 88.2

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

   5. Applied rewrites 88.2%

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

Alternative 6: 88.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\log y, x, \log t\right) - y\\ \mathbf{if}\;x \leq -4.1 \cdot 10^{+114}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+15}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (fma (log y) x (log t)) y)))
   (if (<= x -4.1e+114) t_1 (if (<= x 7.4e+15) (- (- (log t) y) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(log(y), x, log(t)) - y;
	double tmp;
	if (x <= -4.1e+114) {
		tmp = t_1;
	} else if (x <= 7.4e+15) {
		tmp = (log(t) - y) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(log(y), x, log(t)) - y)
	tmp = 0.0
	if (x <= -4.1e+114)
		tmp = t_1;
	elseif (x <= 7.4e+15)
		tmp = Float64(Float64(log(t) - y) - z);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Log[y], $MachinePrecision] * x + N[Log[t], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[x, -4.1e+114], t$95$1, If[LessEqual[x, 7.4e+15], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.1000000000000001e114 or 7.4e15 < 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 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 84.6

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

   5. Applied rewrites 84.6%

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

   if -4.1000000000000001e114 < x < 7.4e15

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate--r+ N/A

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

      2. 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 97.8

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

   5. Applied rewrites 97.8%

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

Alternative 7: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+168}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+109}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -6.5e+168)
   (* (log y) x)
   (if (<= x 1.75e+109) (- (- (log t) y) z) (fma (log y) x (log t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -6.5e+168) {
		tmp = log(y) * x;
	} else if (x <= 1.75e+109) {
		tmp = (log(t) - y) - z;
	} else {
		tmp = fma(log(y), x, log(t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -6.5e+168)
		tmp = Float64(log(y) * x);
	elseif (x <= 1.75e+109)
		tmp = Float64(Float64(log(t) - y) - z);
	else
		tmp = fma(log(y), x, log(t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -6.5e+168], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 1.75e+109], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x + N[Log[t], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.49999999999999999e168

   1. Initial program 99.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 96.5%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} \]
   6. 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 86.4

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

   7. Applied rewrites 86.4%

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

   if -6.49999999999999999e168 < x < 1.74999999999999992e109

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate--r+ N/A

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

      2. 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 90.2

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

   5. Applied rewrites 90.2%

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

   if 1.74999999999999992e109 < 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 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 88.6

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

   5. Applied rewrites 88.6%

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

   7. Applied rewrites 88.6%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 76.7%

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

Alternative 8: 83.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{+168}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+109}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -6.5e+168) t_1 (if (<= x 1.75e+109) (- (- (log t) y) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if (x <= -6.5e+168) {
		tmp = t_1;
	} else if (x <= 1.75e+109) {
		tmp = (log(t) - y) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(y) * x
    if (x <= (-6.5d+168)) then
        tmp = t_1
    else if (x <= 1.75d+109) then
        tmp = (log(t) - y) - z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) * x;
	double tmp;
	if (x <= -6.5e+168) {
		tmp = t_1;
	} else if (x <= 1.75e+109) {
		tmp = (Math.log(t) - y) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	tmp = 0
	if x <= -6.5e+168:
		tmp = t_1
	elif x <= 1.75e+109:
		tmp = (math.log(t) - y) - z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (x <= -6.5e+168)
		tmp = t_1;
	elseif (x <= 1.75e+109)
		tmp = Float64(Float64(log(t) - y) - z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	tmp = 0.0;
	if (x <= -6.5e+168)
		tmp = t_1;
	elseif (x <= 1.75e+109)
		tmp = (log(t) - y) - z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -6.5e+168], t$95$1, If[LessEqual[x, 1.75e+109], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.49999999999999999e168 or 1.74999999999999992e109 < x

   1. Initial program 99.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 97.2%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} \]
   6. 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 80.5

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

   7. Applied rewrites 80.5%

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

   if -6.49999999999999999e168 < x < 1.74999999999999992e109

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate--r+ N/A

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

      2. 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 90.2

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

   5. Applied rewrites 90.2%

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

Alternative 9: 60.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+90}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 0.0132:\\ \;\;\;\;\log t - y\\ \mathbf{else}:\\ \;\;\;\;\log t - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.35e+90) (- z) (if (<= z 0.0132) (- (log t) y) (- (log t) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.35e+90) {
		tmp = -z;
	} else if (z <= 0.0132) {
		tmp = log(t) - y;
	} else {
		tmp = log(t) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1.35d+90)) then
        tmp = -z
    else if (z <= 0.0132d0) then
        tmp = log(t) - y
    else
        tmp = log(t) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.35e+90) {
		tmp = -z;
	} else if (z <= 0.0132) {
		tmp = Math.log(t) - y;
	} else {
		tmp = Math.log(t) - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.35e+90:
		tmp = -z
	elif z <= 0.0132:
		tmp = math.log(t) - y
	else:
		tmp = math.log(t) - z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.35e+90)
		tmp = Float64(-z);
	elseif (z <= 0.0132)
		tmp = Float64(log(t) - y);
	else
		tmp = Float64(log(t) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.35e+90)
		tmp = -z;
	elseif (z <= 0.0132)
		tmp = log(t) - y;
	else
		tmp = log(t) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.35e+90], (-z), If[LessEqual[z, 0.0132], N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.35e90

   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 76.3

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

   5. Applied rewrites 76.3%

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

   if -1.35e90 < z < 0.0132

   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 97.6

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 55.5%

      \[\leadsto \log t - y \]

   if 0.0132 < 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. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. cancel-sign-sub N/A

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

      9. mul-1-neg N/A

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

      10. *-inverses N/A

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

      11. associate-/l* N/A

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

      12. associate-*l/ N/A

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

      13. associate-*r/ N/A

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

      14. sub-neg N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

   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 x around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 76.8

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

   8. Applied rewrites 76.8%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 65.3%

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

Alternative 10: 60.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{+28}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 2.8e+28) (- (log t) z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.8e+28) {
		tmp = log(t) - z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 2.8d+28) then
        tmp = log(t) - z
    else
        tmp = -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.8e+28) {
		tmp = Math.log(t) - z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 2.8e+28:
		tmp = math.log(t) - z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.8e+28)
		tmp = Float64(log(t) - z);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 2.8e+28)
		tmp = log(t) - z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 2.8e+28], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 2.8000000000000001e28

   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. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. cancel-sign-sub N/A

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

      9. mul-1-neg N/A

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

      10. *-inverses N/A

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

      11. associate-/l* N/A

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

      12. associate-*l/ N/A

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

      13. associate-*r/ N/A

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

      14. sub-neg N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

   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 x around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 63.4

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

   8. Applied rewrites 63.4%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 62.6%

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

   if 2.8000000000000001e28 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. cancel-sign-sub N/A

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

      9. mul-1-neg N/A

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

      10. *-inverses N/A

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

      11. associate-/l* N/A

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

      12. associate-*l/ N/A

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

      13. associate-*r/ N/A

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

      14. sub-neg N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

   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 y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 54.0

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

   8. Applied rewrites 54.0%

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

Alternative 11: 47.8% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+90}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 1150000:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.35e+90) (- z) (if (<= z 1150000.0) (- y) (- z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.35e+90) {
		tmp = -z;
	} else if (z <= 1150000.0) {
		tmp = -y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1.35d+90)) then
        tmp = -z
    else if (z <= 1150000.0d0) then
        tmp = -y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.35e+90) {
		tmp = -z;
	} else if (z <= 1150000.0) {
		tmp = -y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.35e+90:
		tmp = -z
	elif z <= 1150000.0:
		tmp = -y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.35e+90)
		tmp = Float64(-z);
	elseif (z <= 1150000.0)
		tmp = Float64(-y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.35e+90)
		tmp = -z;
	elseif (z <= 1150000.0)
		tmp = -y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.35e+90], (-z), If[LessEqual[z, 1150000.0], (-y), (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -1.35e90 or 1.15e6 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 69.8

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

   5. Applied rewrites 69.8%

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

   if -1.35e90 < z < 1.15e6

   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. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. cancel-sign-sub N/A

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

      9. mul-1-neg N/A

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

      10. *-inverses N/A

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

      11. associate-/l* N/A

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

      12. associate-*l/ N/A

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

      13. associate-*r/ N/A

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

      14. sub-neg N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

   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 y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 38.9

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

   8. Applied rewrites 38.9%

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

Alternative 12: 29.8% accurate, 71.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in 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. remove-double-neg N/A

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

   4. mul-1-neg N/A

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

   5. *-commutative N/A

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

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

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

   7. metadata-eval N/A

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

   8. cancel-sign-sub N/A

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

   9. mul-1-neg N/A

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

   10. *-inverses N/A

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

   11. associate-/l* N/A

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

   12. associate-*l/ N/A

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

   13. associate-*r/ N/A

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

   14. sub-neg N/A

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

   15. *-commutative N/A

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

   16. lower-fma.f64 N/A

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

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 y around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 27.6

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

8. Applied rewrites 27.6%

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

Reproduce

herbie shell --seed 2024332 
(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)))