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

Percentage Accurate: 99.9% → 99.9%

Time: 9.2s

Alternatives: 8

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := N[(N[Log[y], $MachinePrecision] * x + N[(N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision] - y), $MachinePrecision]), $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. lift--.f64 N/A

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

   3. lift--.f64 N/A

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

   4. associate--l- N/A

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

   5. associate-+l- N/A

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

   6. sub-neg N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. associate-+r- N/A

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

   12. lower-+.f64 N/A

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

   13. lower--.f64 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-fma.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. unsub-neg N/A

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

   4. lift-+.f64 N/A

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

   5. associate--r+ N/A

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

   6. *-commutative N/A

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

   7. lift-log.f64 N/A

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

   8. lift--.f64 N/A

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

   9. associate-+l- N/A

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

   10. +-commutative N/A

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

   11. associate--l- N/A

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

   12. lift-log.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-+r- N/A

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

   15. +-commutative N/A

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

   16. lift-fma.f64 N/A

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

   17. +-commutative N/A

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

   18. associate--l- N/A

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

6. Applied rewrites 99.8%

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

Alternative 2: 87.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - y\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+236}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -y\right)\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{+43}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) y)))
   (if (<= t_1 -5e+236)
     (fma (log y) x (- y))
     (if (<= t_1 -4e+43) (- (- (log t) y) z) (- (fma (log y) x (log t)) z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - y;
	double tmp;
	if (t_1 <= -5e+236) {
		tmp = fma(log(y), x, -y);
	} else if (t_1 <= -4e+43) {
		tmp = (log(t) - y) - z;
	} else {
		tmp = fma(log(y), x, log(t)) - z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - y)
	tmp = 0.0
	if (t_1 <= -5e+236)
		tmp = fma(log(y), x, Float64(-y));
	elseif (t_1 <= -4e+43)
		tmp = Float64(Float64(log(t) - y) - 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[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+236], N[(N[Log[y], $MachinePrecision] * x + (-y)), $MachinePrecision], If[LessEqual[t$95$1, -4e+43], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - 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) < -4.9999999999999997e236

   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. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate--l- N/A

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

      5. associate-+l- N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. associate-+r- N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 95.8

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

   7. Applied rewrites 95.8%

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

   if -4.9999999999999997e236 < (-.f64 (*.f64 x (log.f64 y)) y) < -4.00000000000000006e43

   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 86.0

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

   5. Applied rewrites 86.0%

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

   if -4.00000000000000006e43 < (-.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 96.8

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

   5. Applied rewrites 96.8%

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

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. associate--r+ N/A

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

   3. associate--l+ N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. mul-1-neg N/A

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

   7. lower--.f64 N/A

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

   8. mul-1-neg N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. associate--l+ N/A

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

   12. lower--.f64 N/A

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

   13. +-commutative N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-log.f64 N/A

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

   17. lower-log.f64 99.8

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

5. Applied rewrites 99.8%

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

Alternative 4: 89.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\log y, x, -y\right)\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{+134}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+70}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (log y) x (- y))))
   (if (<= x -2.3e+134) t_1 (if (<= x 5.2e+70) (- (- (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 <= -2.3e+134) {
		tmp = t_1;
	} else if (x <= 5.2e+70) {
		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 <= -2.3e+134)
		tmp = t_1;
	elseif (x <= 5.2e+70)
		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, -2.3e+134], t$95$1, If[LessEqual[x, 5.2e+70], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2999999999999998e134 or 5.2000000000000001e70 < x

   1. Initial program 99.5%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   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. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate--l- N/A

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

      5. associate-+l- N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. associate-+r- N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 87.1

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

   7. Applied rewrites 87.1%

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

   if -2.2999999999999998e134 < x < 5.2000000000000001e70

   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 94.7

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

   5. Applied rewrites 94.7%

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

Alternative 5: 84.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -2.8 \cdot 10^{+134}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+123}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -2.8e+134) t_1 (if (<= x 7.5e+123) (- (- (log t) y) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -2.8e+134) {
		tmp = t_1;
	} else if (x <= 7.5e+123) {
		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 = x * log(y)
    if (x <= (-2.8d+134)) then
        tmp = t_1
    else if (x <= 7.5d+123) 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 = x * Math.log(y);
	double tmp;
	if (x <= -2.8e+134) {
		tmp = t_1;
	} else if (x <= 7.5e+123) {
		tmp = (Math.log(t) - y) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -2.8e+134:
		tmp = t_1
	elif x <= 7.5e+123:
		tmp = (math.log(t) - y) - z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -2.8e+134)
		tmp = t_1;
	elseif (x <= 7.5e+123)
		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 = x * log(y);
	tmp = 0.0;
	if (x <= -2.8e+134)
		tmp = t_1;
	elseif (x <= 7.5e+123)
		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[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.8e+134], t$95$1, If[LessEqual[x, 7.5e+123], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.7999999999999999e134 or 7.4999999999999999e123 < x

   1. Initial program 99.5%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   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. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate--l- N/A

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

      5. associate-+l- N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. associate-+r- N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 79.7

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

   7. Applied rewrites 79.7%

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

   if -2.7999999999999999e134 < x < 7.4999999999999999e123

   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 93.4

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

   5. Applied rewrites 93.4%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+134}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+123}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+62}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+51}:\\ \;\;\;\;\log t - y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -4e+62) (- z) (if (<= z 1.65e+51) (- (log t) y) (- z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4e+62) {
		tmp = -z;
	} else if (z <= 1.65e+51) {
		tmp = log(t) - 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 <= (-4d+62)) then
        tmp = -z
    else if (z <= 1.65d+51) then
        tmp = log(t) - 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 <= -4e+62) {
		tmp = -z;
	} else if (z <= 1.65e+51) {
		tmp = Math.log(t) - y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -4e+62:
		tmp = -z
	elif z <= 1.65e+51:
		tmp = math.log(t) - y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -4e+62)
		tmp = Float64(-z);
	elseif (z <= 1.65e+51)
		tmp = Float64(log(t) - y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -4e+62)
		tmp = -z;
	elseif (z <= 1.65e+51)
		tmp = log(t) - y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -4e+62], (-z), If[LessEqual[z, 1.65e+51], N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -4.00000000000000014e62 or 1.6499999999999999e51 < 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 65.8

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

   5. Applied rewrites 65.8%

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

   if -4.00000000000000014e62 < z < 1.6499999999999999e51

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 65.7%

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

Alternative 7: 48.5% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+62}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+51}:\\ \;\;\;\;-1 \cdot y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -4e+62) (- z) (if (<= z 1.3e+51) (* -1.0 y) (- z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4e+62) {
		tmp = -z;
	} else if (z <= 1.3e+51) {
		tmp = -1.0 * 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 <= (-4d+62)) then
        tmp = -z
    else if (z <= 1.3d+51) then
        tmp = (-1.0d0) * 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 <= -4e+62) {
		tmp = -z;
	} else if (z <= 1.3e+51) {
		tmp = -1.0 * y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -4e+62:
		tmp = -z
	elif z <= 1.3e+51:
		tmp = -1.0 * y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -4e+62)
		tmp = Float64(-z);
	elseif (z <= 1.3e+51)
		tmp = Float64(-1.0 * y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -4e+62)
		tmp = -z;
	elseif (z <= 1.3e+51)
		tmp = -1.0 * y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -4e+62], (-z), If[LessEqual[z, 1.3e+51], N[(-1.0 * y), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -4.00000000000000014e62 or 1.3000000000000001e51 < 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 65.8

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

   5. Applied rewrites 65.8%

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

   if -4.00000000000000014e62 < z < 1.3000000000000001e51

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 85.5%

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

   9. Taylor expanded in y around inf

      \[\leadsto -1 \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 41.3%

       \[\leadsto -1 \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 29.6% accurate, 71.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
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 29.4

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

5. Applied rewrites 29.4%

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

Reproduce

herbie shell --seed 2024298 
(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)))