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

Percentage Accurate: 99.9% → 99.9%

Time: 10.4s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
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: 70.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ t_2 := \left(t\_1 - y\right) - z\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+14}:\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\left(-y\right) + \log t\\ \mathbf{elif}\;t\_2 \leq 10^{+169}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)) (t_2 (- (- t_1 y) z)))
   (if (<= t_2 -5e+14)
     (- (- z) y)
     (if (<= t_2 5e-5) (+ (- y) (log t)) (if (<= t_2 1e+169) (- 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) - z;
	double tmp;
	if (t_2 <= -5e+14) {
		tmp = -z - y;
	} else if (t_2 <= 5e-5) {
		tmp = -y + log(t);
	} else if (t_2 <= 1e+169) {
		tmp = -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) - z
    if (t_2 <= (-5d+14)) then
        tmp = -z - y
    else if (t_2 <= 5d-5) then
        tmp = -y + log(t)
    else if (t_2 <= 1d+169) then
        tmp = -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) - z;
	double tmp;
	if (t_2 <= -5e+14) {
		tmp = -z - y;
	} else if (t_2 <= 5e-5) {
		tmp = -y + Math.log(t);
	} else if (t_2 <= 1e+169) {
		tmp = -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) - z
	tmp = 0
	if t_2 <= -5e+14:
		tmp = -z - y
	elif t_2 <= 5e-5:
		tmp = -y + math.log(t)
	elif t_2 <= 1e+169:
		tmp = -z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	t_2 = Float64(Float64(t_1 - y) - z)
	tmp = 0.0
	if (t_2 <= -5e+14)
		tmp = Float64(Float64(-z) - y);
	elseif (t_2 <= 5e-5)
		tmp = Float64(Float64(-y) + log(t));
	elseif (t_2 <= 1e+169)
		tmp = Float64(-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) - z;
	tmp = 0.0;
	if (t_2 <= -5e+14)
		tmp = -z - y;
	elseif (t_2 <= 5e-5)
		tmp = -y + log(t);
	elseif (t_2 <= 1e+169)
		tmp = -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[(N[(t$95$1 - y), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+14], N[((-z) - y), $MachinePrecision], If[LessEqual[t$95$2, 5e-5], N[((-y) + N[Log[t], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+169], (-z), t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) < -5e14

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 76.4

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

   7. Applied rewrites 76.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. remove-double-div 76.5

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

   9. Applied rewrites 76.5%

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

   if -5e14 < (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) < 5.00000000000000024e-5

   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 inf

      \[\leadsto \color{blue}{-1 \cdot y} + \log t \]
   4. 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 94.9

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

   5. Applied rewrites 94.9%

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

   if 5.00000000000000024e-5 < (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) < 9.99999999999999934e168

   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 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 61.4

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

   5. Applied rewrites 61.4%

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

   if 9.99999999999999934e168 < (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z)

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} \]
   4. 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 69.2

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

   5. Applied rewrites 69.2%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\log y \cdot x - y\right) - z \leq -5 \cdot 10^{+14}:\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{elif}\;\left(\log y \cdot x - y\right) - z \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\left(-y\right) + \log t\\ \mathbf{elif}\;\left(\log y \cdot x - y\right) - z \leq 10^{+169}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x - y\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+156}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -y\right)\\ \mathbf{elif}\;t\_1 \leq -4000000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{\log y}{z}, x, \frac{-y}{z} - 1\right) \cdot z\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (log y) x) y)))
   (if (<= t_1 -4e+156)
     (fma (log y) x (- y))
     (if (<= t_1 -4000000000000.0)
       (* (fma (/ (log y) z) x (- (/ (- y) z) 1.0)) z)
       (if (<= t_1 5e-8) (- (- (log t) y) z) (fma (log y) x (- z)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (log(y) * x) - y;
	double tmp;
	if (t_1 <= -4e+156) {
		tmp = fma(log(y), x, -y);
	} else if (t_1 <= -4000000000000.0) {
		tmp = fma((log(y) / z), x, ((-y / z) - 1.0)) * z;
	} else if (t_1 <= 5e-8) {
		tmp = (log(t) - y) - z;
	} else {
		tmp = fma(log(y), x, -z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(log(y) * x) - y)
	tmp = 0.0
	if (t_1 <= -4e+156)
		tmp = fma(log(y), x, Float64(-y));
	elseif (t_1 <= -4000000000000.0)
		tmp = Float64(fma(Float64(log(y) / z), x, Float64(Float64(Float64(-y) / z) - 1.0)) * z);
	elseif (t_1 <= 5e-8)
		tmp = Float64(Float64(log(t) - y) - z);
	else
		tmp = fma(log(y), x, Float64(-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]}, If[LessEqual[t$95$1, -4e+156], N[(N[Log[y], $MachinePrecision] * x + (-y)), $MachinePrecision], If[LessEqual[t$95$1, -4000000000000.0], N[(N[(N[(N[Log[y], $MachinePrecision] / z), $MachinePrecision] * x + N[(N[((-y) / z), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$1, 5e-8], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x + (-z)), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -3.9999999999999999e156

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. cancel-sign-sub N/A

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

      9. mul-1-neg N/A

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

      10. *-inverses N/A

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

      11. associate-/l* N/A

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

      12. associate-*l/ N/A

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

      13. associate-*r/ N/A

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

      14. sub-neg N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 91.4%

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

   if -3.9999999999999999e156 < (-.f64 (*.f64 x (log.f64 y)) y) < -4e12

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 94.2%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 93.9%

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

   if -4e12 < (-.f64 (*.f64 x (log.f64 y)) y) < 4.9999999999999998e-8

   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 98.4

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

   5. Applied rewrites 98.4%

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

   if 4.9999999999999998e-8 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. 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 z around inf

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

   8. Applied rewrites 97.5%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log y \cdot x - y \leq -4 \cdot 10^{+156}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -y\right)\\ \mathbf{elif}\;\log y \cdot x - y \leq -4000000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{\log y}{z}, x, \frac{-y}{z} - 1\right) \cdot z\\ \mathbf{elif}\;\log y \cdot x - y \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x - y\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+168}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -y\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (log y) x) y)))
   (if (<= t_1 -2e+168)
     (fma (log y) x (- y))
     (if (<= t_1 5e-8) (- (- (log t) y) z) (fma (log y) x (- z))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (log(y) * x) - y;
	double tmp;
	if (t_1 <= -2e+168) {
		tmp = fma(log(y), x, -y);
	} else if (t_1 <= 5e-8) {
		tmp = (log(t) - y) - z;
	} else {
		tmp = fma(log(y), x, -z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(log(y) * x) - y)
	tmp = 0.0
	if (t_1 <= -2e+168)
		tmp = fma(log(y), x, Float64(-y));
	elseif (t_1 <= 5e-8)
		tmp = Float64(Float64(log(t) - y) - z);
	else
		tmp = fma(log(y), x, Float64(-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]}, If[LessEqual[t$95$1, -2e+168], N[(N[Log[y], $MachinePrecision] * x + (-y)), $MachinePrecision], If[LessEqual[t$95$1, 5e-8], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x + (-z)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -1.9999999999999999e168

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. cancel-sign-sub N/A

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

      9. mul-1-neg N/A

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

      10. *-inverses N/A

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

      11. associate-/l* N/A

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

      12. associate-*l/ N/A

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

      13. associate-*r/ N/A

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

      14. sub-neg N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 93.3%

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

   if -1.9999999999999999e168 < (-.f64 (*.f64 x (log.f64 y)) y) < 4.9999999999999998e-8

   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 91.3

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

   5. Applied rewrites 91.3%

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

   if 4.9999999999999998e-8 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. 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 z around inf

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

   8. Applied rewrites 97.5%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log y \cdot x - y \leq -2 \cdot 10^{+168}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -y\right)\\ \mathbf{elif}\;\log y \cdot x - y \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ t_2 := t\_1 - y\\ \mathbf{if}\;t\_2 \leq -50000000:\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+42}:\\ \;\;\;\;\left(-z\right) + \log t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)) (t_2 (- t_1 y)))
   (if (<= t_2 -50000000.0)
     (- (- z) y)
     (if (<= t_2 2e+42) (+ (- z) (log t)) 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 <= -50000000.0) {
		tmp = -z - y;
	} else if (t_2 <= 2e+42) {
		tmp = -z + log(t);
	} 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 <= (-50000000.0d0)) then
        tmp = -z - y
    else if (t_2 <= 2d+42) then
        tmp = -z + log(t)
    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 <= -50000000.0) {
		tmp = -z - y;
	} else if (t_2 <= 2e+42) {
		tmp = -z + Math.log(t);
	} 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 <= -50000000.0:
		tmp = -z - y
	elif t_2 <= 2e+42:
		tmp = -z + math.log(t)
	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 <= -50000000.0)
		tmp = Float64(Float64(-z) - y);
	elseif (t_2 <= 2e+42)
		tmp = Float64(Float64(-z) + log(t));
	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 <= -50000000.0)
		tmp = -z - y;
	elseif (t_2 <= 2e+42)
		tmp = -z + log(t);
	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, -50000000.0], N[((-z) - y), $MachinePrecision], If[LessEqual[t$95$2, 2e+42], N[((-z) + N[Log[t], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -5e7

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 76.0

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

   7. Applied rewrites 76.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. remove-double-div 76.1

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

   9. Applied rewrites 76.1%

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

   if -5e7 < (-.f64 (*.f64 x (log.f64 y)) y) < 2.00000000000000009e42

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 94.4

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

   5. Applied rewrites 94.4%

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

   if 2.00000000000000009e42 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} \]
   4. 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 83.5

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

   5. Applied rewrites 83.5%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log y \cdot x - y \leq -50000000:\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{elif}\;\log y \cdot x - y \leq 2 \cdot 10^{+42}:\\ \;\;\;\;\left(-z\right) + \log t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{\log y}{z}, x, \frac{-y}{z} - 1\right) \cdot z\\ \mathbf{if}\;z \leq -1400:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 520:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t\right) - y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma (/ (log y) z) x (- (/ (- y) z) 1.0)) z)))
   (if (<= z -1400.0)
     t_1
     (if (<= z 520.0) (- (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) / z), x, ((-y / z) - 1.0)) * z;
	double tmp;
	if (z <= -1400.0) {
		tmp = t_1;
	} else if (z <= 520.0) {
		tmp = fma(log(y), x, log(t)) - y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(Float64(log(y) / z), x, Float64(Float64(Float64(-y) / z) - 1.0)) * z)
	tmp = 0.0
	if (z <= -1400.0)
		tmp = t_1;
	elseif (z <= 520.0)
		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] / z), $MachinePrecision] * x + N[(N[((-y) / z), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -1400.0], t$95$1, If[LessEqual[z, 520.0], 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 < -1400 or 520 < z

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 98.9%

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

   if -1400 < z < 520

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 99.0

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

   5. Applied rewrites 99.0%

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

Alternative 7: 90.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\log y, x, -y\right)\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{+36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+62}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (log y) x (- y))))
   (if (<= x -1.1e+36) t_1 (if (<= x 1.65e+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, -y);
	double tmp;
	if (x <= -1.1e+36) {
		tmp = t_1;
	} else if (x <= 1.65e+62) {
		tmp = (log(t) - y) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(log(y), x, Float64(-y))
	tmp = 0.0
	if (x <= -1.1e+36)
		tmp = t_1;
	elseif (x <= 1.65e+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[Log[y], $MachinePrecision] * x + (-y)), $MachinePrecision]}, If[LessEqual[x, -1.1e+36], t$95$1, If[LessEqual[x, 1.65e+62], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1e36 or 1.65e62 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. 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 99.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 85.3%

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

   if -1.1e36 < x < 1.65e62

   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 97.4

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

   5. Applied rewrites 97.4%

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

Alternative 8: 84.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{+166}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+162}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -6.5e+166) t_1 (if (<= x 4.6e+162) (- (- (log t) y) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if (x <= -6.5e+166) {
		tmp = t_1;
	} else if (x <= 4.6e+162) {
		tmp = (log(t) - y) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(y) * x
    if (x <= (-6.5d+166)) then
        tmp = t_1
    else if (x <= 4.6d+162) then
        tmp = (log(t) - y) - z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) * x;
	double tmp;
	if (x <= -6.5e+166) {
		tmp = t_1;
	} else if (x <= 4.6e+162) {
		tmp = (Math.log(t) - y) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	tmp = 0
	if x <= -6.5e+166:
		tmp = t_1
	elif x <= 4.6e+162:
		tmp = (math.log(t) - y) - z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (x <= -6.5e+166)
		tmp = t_1;
	elseif (x <= 4.6e+162)
		tmp = Float64(Float64(log(t) - y) - z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	tmp = 0.0;
	if (x <= -6.5e+166)
		tmp = t_1;
	elseif (x <= 4.6e+162)
		tmp = (log(t) - y) - z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -6.5e+166], t$95$1, If[LessEqual[x, 4.6e+162], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.5000000000000005e166 or 4.59999999999999987e162 < 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 inf

      \[\leadsto \color{blue}{x \cdot \log y} \]
   4. 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 85.7

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

   5. Applied rewrites 85.7%

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

   if -6.5000000000000005e166 < x < 4.59999999999999987e162

   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 88.6

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

   5. Applied rewrites 88.6%

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

Alternative 9: 71.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{+166}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+162}:\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -6.5e+166) t_1 (if (<= x 4.6e+162) (- (- z) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if (x <= -6.5e+166) {
		tmp = t_1;
	} else if (x <= 4.6e+162) {
		tmp = -z - y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(y) * x
    if (x <= (-6.5d+166)) then
        tmp = t_1
    else if (x <= 4.6d+162) then
        tmp = -z - y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) * x;
	double tmp;
	if (x <= -6.5e+166) {
		tmp = t_1;
	} else if (x <= 4.6e+162) {
		tmp = -z - y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	tmp = 0
	if x <= -6.5e+166:
		tmp = t_1
	elif x <= 4.6e+162:
		tmp = -z - y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (x <= -6.5e+166)
		tmp = t_1;
	elseif (x <= 4.6e+162)
		tmp = Float64(Float64(-z) - y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	tmp = 0.0;
	if (x <= -6.5e+166)
		tmp = t_1;
	elseif (x <= 4.6e+162)
		tmp = -z - y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -6.5e+166], t$95$1, If[LessEqual[x, 4.6e+162], N[((-z) - y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.5000000000000005e166 or 4.59999999999999987e162 < 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 inf

      \[\leadsto \color{blue}{x \cdot \log y} \]
   4. 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 85.7

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

   5. Applied rewrites 85.7%

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

   if -6.5000000000000005e166 < x < 4.59999999999999987e162

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 68.6

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

   7. Applied rewrites 68.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. remove-double-div 68.7

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

   9. Applied rewrites 68.7%

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

Alternative 10: 48.3% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.3 \cdot 10^{+22}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+117}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -6.3e+22) (- z) (if (<= z 1.2e+117) (- y) (- z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -6.3e+22) {
		tmp = -z;
	} else if (z <= 1.2e+117) {
		tmp = -y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-6.3d+22)) then
        tmp = -z
    else if (z <= 1.2d+117) then
        tmp = -y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -6.3e+22) {
		tmp = -z;
	} else if (z <= 1.2e+117) {
		tmp = -y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -6.3e+22:
		tmp = -z
	elif z <= 1.2e+117:
		tmp = -y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -6.3e+22)
		tmp = Float64(-z);
	elseif (z <= 1.2e+117)
		tmp = Float64(-y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -6.3e+22)
		tmp = -z;
	elseif (z <= 1.2e+117)
		tmp = -y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -6.3e+22], (-z), If[LessEqual[z, 1.2e+117], (-y), (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -6.30000000000000021e22 or 1.1999999999999999e117 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 66.0

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

   5. Applied rewrites 66.0%

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

   if -6.30000000000000021e22 < z < 1.1999999999999999e117

   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 inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 39.6

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

   5. Applied rewrites 39.6%

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

Alternative 11: 58.2% accurate, 35.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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

   2. flip3-+ N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip3-+ N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 55.4

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

7. Applied rewrites 55.4%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. remove-double-div 55.5

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

9. Applied rewrites 55.5%

   \[\leadsto \color{blue}{\left(-z\right) - y} \]
10. Add Preprocessing

Alternative 12: 30.7% 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. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 30.3

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

5. Applied rewrites 30.3%

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

Reproduce

herbie shell --seed 2024235 
(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)))