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

Percentage Accurate: 99.9% → 99.9%

Time: 9.2s

Alternatives: 13

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 99.8%

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

Alternative 2: 64.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ t_2 := t\_1 - y\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+187}:\\ \;\;\;\;\left(-y\right) + \log t\\ \mathbf{elif}\;t\_2 \leq -3 \cdot 10^{+163}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+104}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)) (t_2 (- t_1 y)))
   (if (<= t_2 -5e+187)
     (+ (- y) (log t))
     (if (<= t_2 -3e+163) t_1 (if (<= t_2 2e+104) (- (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 <= -5e+187) {
		tmp = -y + log(t);
	} else if (t_2 <= -3e+163) {
		tmp = t_1;
	} else if (t_2 <= 2e+104) {
		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 <= (-5d+187)) then
        tmp = -y + log(t)
    else if (t_2 <= (-3d+163)) then
        tmp = t_1
    else if (t_2 <= 2d+104) 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 <= -5e+187) {
		tmp = -y + Math.log(t);
	} else if (t_2 <= -3e+163) {
		tmp = t_1;
	} else if (t_2 <= 2e+104) {
		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 <= -5e+187:
		tmp = -y + math.log(t)
	elif t_2 <= -3e+163:
		tmp = t_1
	elif t_2 <= 2e+104:
		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 <= -5e+187)
		tmp = Float64(Float64(-y) + log(t));
	elseif (t_2 <= -3e+163)
		tmp = t_1;
	elseif (t_2 <= 2e+104)
		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 <= -5e+187)
		tmp = -y + log(t);
	elseif (t_2 <= -3e+163)
		tmp = t_1;
	elseif (t_2 <= 2e+104)
		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, -5e+187], N[((-y) + N[Log[t], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -3e+163], t$95$1, If[LessEqual[t$95$2, 2e+104], 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) < -5.0000000000000001e187

   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

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

      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. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. inv-pow N/A

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

      6. lower-pow.f64 99.8

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.8

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in y around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 71.0

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

   11. Applied rewrites 71.0%

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

   if -5.0000000000000001e187 < (-.f64 (*.f64 x (log.f64 y)) y) < -3.00000000000000013e163 or 2e104 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.5%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   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 99.5%

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

   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 72.3

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

   8. Applied rewrites 72.3%

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

   if -3.00000000000000013e163 < (-.f64 (*.f64 x (log.f64 y)) y) < 2e104

   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 81.0

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

   5. Applied rewrites 81.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 71.7%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log y \cdot x - y \leq -5 \cdot 10^{+187}:\\ \;\;\;\;\left(-y\right) + \log t\\ \mathbf{elif}\;\log y \cdot x - y \leq -3 \cdot 10^{+163}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;\log y \cdot x - y \leq 2 \cdot 10^{+104}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ t_2 := t\_1 - y\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+187}:\\ \;\;\;\;-y\\ \mathbf{elif}\;t\_2 \leq -3 \cdot 10^{+163}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+104}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)) (t_2 (- t_1 y)))
   (if (<= t_2 -5e+187)
     (- y)
     (if (<= t_2 -3e+163) t_1 (if (<= t_2 2e+104) (- (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 <= -5e+187) {
		tmp = -y;
	} else if (t_2 <= -3e+163) {
		tmp = t_1;
	} else if (t_2 <= 2e+104) {
		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 <= (-5d+187)) then
        tmp = -y
    else if (t_2 <= (-3d+163)) then
        tmp = t_1
    else if (t_2 <= 2d+104) 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 <= -5e+187) {
		tmp = -y;
	} else if (t_2 <= -3e+163) {
		tmp = t_1;
	} else if (t_2 <= 2e+104) {
		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 <= -5e+187:
		tmp = -y
	elif t_2 <= -3e+163:
		tmp = t_1
	elif t_2 <= 2e+104:
		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 <= -5e+187)
		tmp = Float64(-y);
	elseif (t_2 <= -3e+163)
		tmp = t_1;
	elseif (t_2 <= 2e+104)
		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 <= -5e+187)
		tmp = -y;
	elseif (t_2 <= -3e+163)
		tmp = t_1;
	elseif (t_2 <= 2e+104)
		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, -5e+187], (-y), If[LessEqual[t$95$2, -3e+163], t$95$1, If[LessEqual[t$95$2, 2e+104], 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) < -5.0000000000000001e187

   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

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

      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. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 70.9

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

   9. Applied rewrites 70.9%

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

   if -5.0000000000000001e187 < (-.f64 (*.f64 x (log.f64 y)) y) < -3.00000000000000013e163 or 2e104 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.5%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   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 99.5%

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

   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 72.3

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

   8. Applied rewrites 72.3%

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

   if -3.00000000000000013e163 < (-.f64 (*.f64 x (log.f64 y)) y) < 2e104

   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 81.0

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

   5. Applied rewrites 81.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 71.7%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log y \cdot x - y \leq -5 \cdot 10^{+187}:\\ \;\;\;\;-y\\ \mathbf{elif}\;\log y \cdot x - y \leq -3 \cdot 10^{+163}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;\log y \cdot x - y \leq 2 \cdot 10^{+104}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x - y\\ t_2 := \log t - y\\ \mathbf{if}\;t\_1 \leq -3 \cdot 10^{+163}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, t\_2\right)\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-7}:\\ \;\;\;\;t\_2 - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (log y) x) y)) (t_2 (- (log t) y)))
   (if (<= t_1 -3e+163)
     (fma (log y) x t_2)
     (if (<= t_1 -2e-7) (- t_2 z) (fma (log y) x (- (log t) z))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (log(y) * x) - y;
	double t_2 = log(t) - y;
	double tmp;
	if (t_1 <= -3e+163) {
		tmp = fma(log(y), x, t_2);
	} else if (t_1 <= -2e-7) {
		tmp = t_2 - z;
	} else {
		tmp = fma(log(y), x, (log(t) - z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(log(y) * x) - y)
	t_2 = Float64(log(t) - y)
	tmp = 0.0
	if (t_1 <= -3e+163)
		tmp = fma(log(y), x, t_2);
	elseif (t_1 <= -2e-7)
		tmp = Float64(t_2 - z);
	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[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - y), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t$95$1, -3e+163], N[(N[Log[y], $MachinePrecision] * x + t$95$2), $MachinePrecision], If[LessEqual[t$95$1, -2e-7], N[(t$95$2 - z), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x + N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -3.00000000000000013e163

   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 z around 0

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

   8. Applied rewrites 92.0%

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

   if -3.00000000000000013e163 < (-.f64 (*.f64 x (log.f64 y)) y) < -1.9999999999999999e-7

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 88.3

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

   5. Applied rewrites 88.3%

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

   if -1.9999999999999999e-7 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. 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 98.9

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

   5. Applied rewrites 98.9%

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

   7. Applied rewrites 98.9%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log y \cdot x - y \leq -3 \cdot 10^{+163}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t - y\right)\\ \mathbf{elif}\;\log y \cdot x - y \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x - y\\ t_2 := \log t - y\\ \mathbf{if}\;t\_1 \leq -3 \cdot 10^{+163}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, t\_2\right)\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-7}:\\ \;\;\;\;t\_2 - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (log y) x) y)) (t_2 (- (log t) y)))
   (if (<= t_1 -3e+163)
     (fma (log y) x t_2)
     (if (<= t_1 -2e-7) (- t_2 z) (- (fma (log y) x (log t)) z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (log(y) * x) - y;
	double t_2 = log(t) - y;
	double tmp;
	if (t_1 <= -3e+163) {
		tmp = fma(log(y), x, t_2);
	} else if (t_1 <= -2e-7) {
		tmp = t_2 - z;
	} else {
		tmp = fma(log(y), x, log(t)) - z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(log(y) * x) - y)
	t_2 = Float64(log(t) - y)
	tmp = 0.0
	if (t_1 <= -3e+163)
		tmp = fma(log(y), x, t_2);
	elseif (t_1 <= -2e-7)
		tmp = Float64(t_2 - z);
	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[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - y), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t$95$1, -3e+163], N[(N[Log[y], $MachinePrecision] * x + t$95$2), $MachinePrecision], If[LessEqual[t$95$1, -2e-7], N[(t$95$2 - z), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * x + N[Log[t], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -3.00000000000000013e163

   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 z around 0

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

   8. Applied rewrites 92.0%

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

   if -3.00000000000000013e163 < (-.f64 (*.f64 x (log.f64 y)) y) < -1.9999999999999999e-7

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 88.3

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

   5. Applied rewrites 88.3%

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

   if -1.9999999999999999e-7 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. 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 98.9

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

   5. Applied rewrites 98.9%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log y \cdot x - y \leq -3 \cdot 10^{+163}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t - y\right)\\ \mathbf{elif}\;\log y \cdot x - y \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x - y\\ t_2 := \mathsf{fma}\left(\log y, x, \log t\right)\\ \mathbf{if}\;t\_1 \leq -3 \cdot 10^{+163}:\\ \;\;\;\;t\_2 - y\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;t\_2 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (log y) x) y)) (t_2 (fma (log y) x (log t))))
   (if (<= t_1 -3e+163)
     (- t_2 y)
     (if (<= t_1 -2e-7) (- (- (log t) y) z) (- t_2 z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (log(y) * x) - y;
	double t_2 = fma(log(y), x, log(t));
	double tmp;
	if (t_1 <= -3e+163) {
		tmp = t_2 - y;
	} else if (t_1 <= -2e-7) {
		tmp = (log(t) - y) - z;
	} else {
		tmp = t_2 - z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(log(y) * x) - y)
	t_2 = fma(log(y), x, log(t))
	tmp = 0.0
	if (t_1 <= -3e+163)
		tmp = Float64(t_2 - y);
	elseif (t_1 <= -2e-7)
		tmp = Float64(Float64(log(t) - y) - z);
	else
		tmp = Float64(t_2 - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - y), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[y], $MachinePrecision] * x + N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -3e+163], N[(t$95$2 - y), $MachinePrecision], If[LessEqual[t$95$1, -2e-7], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], N[(t$95$2 - z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -3.00000000000000013e163

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 92.0

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

   5. Applied rewrites 92.0%

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

   if -3.00000000000000013e163 < (-.f64 (*.f64 x (log.f64 y)) y) < -1.9999999999999999e-7

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 88.3

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

   5. Applied rewrites 88.3%

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

   if -1.9999999999999999e-7 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. 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 98.9

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

   5. Applied rewrites 98.9%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log y \cdot x - y \leq -3 \cdot 10^{+163}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t\right) - y\\ \mathbf{elif}\;\log y \cdot x - y \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{\log y \cdot x}{z}, z, -z\right)\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{+75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t\right) - y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ (* (log y) x) z) z (- z))))
   (if (<= z -1.3e+75)
     t_1
     (if (<= z 8.4e+45) (- (fma (log y) x (log t)) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(((log(y) * x) / z), z, -z);
	double tmp;
	if (z <= -1.3e+75) {
		tmp = t_1;
	} else if (z <= 8.4e+45) {
		tmp = fma(log(y), x, log(t)) - y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(Float64(log(y) * x) / z), z, Float64(-z))
	tmp = 0.0
	if (z <= -1.3e+75)
		tmp = t_1;
	elseif (z <= 8.4e+45)
		tmp = Float64(fma(log(y), x, log(t)) - y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision] * z + (-z)), $MachinePrecision]}, If[LessEqual[z, -1.3e+75], t$95$1, If[LessEqual[z, 8.4e+45], N[(N[(N[Log[y], $MachinePrecision] * x + N[Log[t], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.29999999999999992e75 or 8.39999999999999979e45 < 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 86.5

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

   5. Applied rewrites 86.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 86.4%

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

   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 86.5%

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

   if -1.29999999999999992e75 < z < 8.39999999999999979e45

   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 94.2

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

   5. Applied rewrites 94.2%

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

Alternative 8: 80.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{\log y \cdot x}{z}, z, -z\right)\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{+106}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+62}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ (* (log y) x) z) z (- z))))
   (if (<= x -1.7e+106) t_1 (if (<= x 1.75e+62) (- (- (log t) y) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(((log(y) * x) / z), z, -z);
	double tmp;
	if (x <= -1.7e+106) {
		tmp = t_1;
	} else if (x <= 1.75e+62) {
		tmp = (log(t) - y) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(Float64(log(y) * x) / z), z, Float64(-z))
	tmp = 0.0
	if (x <= -1.7e+106)
		tmp = t_1;
	elseif (x <= 1.75e+62)
		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[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision] * z + (-z)), $MachinePrecision]}, If[LessEqual[x, -1.7e+106], t$95$1, If[LessEqual[x, 1.75e+62], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.69999999999999997e106 or 1.74999999999999992e62 < x

   1. Initial program 99.6%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   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 84.6

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 68.3%

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

   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 68.3%

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

   if -1.69999999999999997e106 < x < 1.74999999999999992e62

   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 97.9

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

   5. Applied rewrites 97.9%

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

Alternative 9: 84.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -9.2 \cdot 10^{+117}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{+157}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -9.2e+117) t_1 (if (<= x 2.65e+157) (- (- (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 <= -9.2e+117) {
		tmp = t_1;
	} else if (x <= 2.65e+157) {
		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 <= (-9.2d+117)) then
        tmp = t_1
    else if (x <= 2.65d+157) 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 <= -9.2e+117) {
		tmp = t_1;
	} else if (x <= 2.65e+157) {
		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 <= -9.2e+117:
		tmp = t_1
	elif x <= 2.65e+157:
		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 <= -9.2e+117)
		tmp = t_1;
	elseif (x <= 2.65e+157)
		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 <= -9.2e+117)
		tmp = t_1;
	elseif (x <= 2.65e+157)
		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, -9.2e+117], t$95$1, If[LessEqual[x, 2.65e+157], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.19999999999999951e117 or 2.6499999999999999e157 < x

   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. 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 99.6%

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

   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 69.4

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

   8. Applied rewrites 69.4%

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

   if -9.19999999999999951e117 < x < 2.6499999999999999e157

   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 91.4

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

   5. Applied rewrites 91.4%

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

Alternative 10: 59.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+108}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 9.5e+108) (- (log t) z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 9.5e+108) {
		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 <= 9.5d+108) 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 <= 9.5e+108) {
		tmp = Math.log(t) - z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 9.5e+108:
		tmp = math.log(t) - z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 9.5e+108)
		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 <= 9.5e+108)
		tmp = log(t) - z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 9.5e+108], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 9.50000000000000097e108

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. 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 90.2

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 58.7%

      \[\leadsto \log t - z \]

   if 9.50000000000000097e108 < y

   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

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

      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. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 98.7%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 71.2

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

   9. Applied rewrites 71.2%

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

Alternative 11: 48.2% accurate, 23.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+108}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 9.5e+108) (- z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 9.5e+108) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 9.5d+108) then
        tmp = -z
    else
        tmp = -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 9.5e+108) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 9.5e+108:
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 9.5e+108)
		tmp = Float64(-z);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 9.5e+108)
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 9.5e+108], (-z), (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 9.50000000000000097e108

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 44.9

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

   5. Applied rewrites 44.9%

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

   if 9.50000000000000097e108 < y

   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

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

      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. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 98.7%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 71.2

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

   9. Applied rewrites 71.2%

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

Alternative 12: 30.5% 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 99.8%

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

   1. lift--.f64 N/A

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

   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. *-commutative N/A

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

   9. associate-/l* N/A

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

   10. lower-fma.f64 N/A

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

4. Applied rewrites 99.1%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate--l+ N/A

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

   4. *-commutative N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. associate-*l* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

6. Applied rewrites 99.3%

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

7. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 29.2

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

9. Applied rewrites 29.2%

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

Alternative 13: 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 99.8%

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

3. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 35.0

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

5. Applied rewrites 35.0%

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

7. Applied rewrites 16.5%

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

9. Applied rewrites 2.1%

   \[\leadsto z \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024294 
(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)))