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

Percentage Accurate: 99.9% → 99.9%

Time: 12.6s

Alternatives: 9

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, 0.7× speedup

Math

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

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(log(t) - Float64(y + z)))
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0 99.8%

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

   1. associate--l+ 99.8%

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

   2. +-commutative 99.8%

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

   3. associate--l- 99.8%

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

   4. fma-def 99.8%

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

   5. associate--l- 99.8%

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

   6. +-commutative 99.8%

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

4. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-+l- 99.8%

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

   2. associate--l- 99.8%

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

   3. +-commutative 99.8%

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

   4. associate--r+ 99.8%

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

   5. fma-neg 99.8%

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

   6. neg-sub0 99.8%

      \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{0 - \left(z - \log t\right)}\right) - y \]

   7. associate-+l- 99.8%

      \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{\left(0 - z\right) + \log t}\right) - y \]

   8. neg-sub0 99.8%

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

   9. +-commutative 99.8%

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

   10. unsub-neg 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[Log[t], $MachinePrecision] + N[(N[(N[(N[Log[y], $MachinePrecision] * x), $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. Final simplification 99.8%

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

Alternative 4: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+36} \lor \neg \left(x \leq 2.7 \cdot 10^{+62}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -5.5e+36) (not (<= x 2.7e+62)))
   (fma (log y) x (- y))
   (- (- (log t) z) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -5.5e+36) || !(x <= 2.7e+62)) {
		tmp = fma(log(y), x, -y);
	} else {
		tmp = (log(t) - z) - y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -5.5e+36) || !(x <= 2.7e+62))
		tmp = fma(log(y), x, Float64(-y));
	else
		tmp = Float64(Float64(log(t) - z) - y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -5.5e+36], N[Not[LessEqual[x, 2.7e+62]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x + (-y)), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5000000000000002e36 or 2.7e62 < x

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 99.6%

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

      1. associate--l+ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate--l- 99.6%

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

      4. fma-def 99.7%

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

      5. associate--l- 99.7%

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

      6. +-commutative 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in y around inf 82.1%

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

      1. neg-mul-1 82.1%

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

   7. Simplified 82.1%

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

   if -5.5000000000000002e36 < x < 2.7e62

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 97.7%

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

      1. +-commutative 97.7%

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

      2. associate--l- 97.7%

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

   4. Simplified 97.7%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+36} \lor \neg \left(x \leq 2.7 \cdot 10^{+62}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \end{array} \]

Alternative 5: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -y\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+42}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -3.9e+36)
   (fma (log y) x (- y))
   (if (<= x 1.35e+42) (- (- (log t) z) y) (fma (log y) x (- z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3.9e+36) {
		tmp = fma(log(y), x, -y);
	} else if (x <= 1.35e+42) {
		tmp = (log(t) - z) - y;
	} else {
		tmp = fma(log(y), x, -z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -3.9e+36)
		tmp = fma(log(y), x, Float64(-y));
	elseif (x <= 1.35e+42)
		tmp = Float64(Float64(log(t) - z) - y);
	else
		tmp = fma(log(y), x, Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -3.9e+36], N[(N[Log[y], $MachinePrecision] * x + (-y)), $MachinePrecision], If[LessEqual[x, 1.35e+42], N[(N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision] - y), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x + (-z)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.90000000000000021e36

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 99.7%

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

      1. associate--l+ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate--l- 99.7%

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

      4. fma-def 99.7%

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

      5. associate--l- 99.7%

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

      6. +-commutative 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in y around inf 80.7%

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

      1. neg-mul-1 80.7%

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

   7. Simplified 80.7%

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

   if -3.90000000000000021e36 < x < 1.35e42

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.0%

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

      1. +-commutative 98.0%

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

      2. associate--l- 98.0%

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

   4. Simplified 98.0%

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

   if 1.35e42 < x

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 99.6%

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

      1. associate--l+ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate--l- 99.6%

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

      4. fma-def 99.6%

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

      5. associate--l- 99.6%

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

      6. +-commutative 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in z around inf 91.5%

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

      1. mul-1-neg 91.5%

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

   7. Simplified 91.5%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -y\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+42}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -z\right)\\ \end{array} \]

Alternative 6: 61.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{+39}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;\log t - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 4.3e+39) (- (log t) z) (- (log t) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.3e+39) {
		tmp = log(t) - z;
	} else {
		tmp = log(t) - y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 4.3d+39) then
        tmp = log(t) - z
    else
        tmp = log(t) - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.3e+39) {
		tmp = Math.log(t) - z;
	} else {
		tmp = Math.log(t) - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 4.3e+39:
		tmp = math.log(t) - z
	else:
		tmp = math.log(t) - y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 4.3e+39)
		tmp = Float64(log(t) - z);
	else
		tmp = Float64(log(t) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 4.3e+39)
		tmp = log(t) - z;
	else
		tmp = log(t) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 4.3e+39], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.3e39

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf 61.7%

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

      1. neg-mul-1 61.7%

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

   4. Simplified 61.7%

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

   if 4.3e39 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 78.2%

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

      1. +-commutative 78.2%

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

      2. associate--l- 78.2%

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

   4. Simplified 78.2%

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

   5. Taylor expanded in z around 0 64.4%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{+39}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;\log t - y\\ \end{array} \]

Alternative 7: 71.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0 71.0%

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

   1. +-commutative 71.0%

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

   2. associate--l- 71.0%

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

4. Simplified 71.0%

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

5. Final simplification 71.0%

   \[\leadsto \left(\log t - z\right) - y \]

Alternative 8: 41.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0 71.0%

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

   1. +-commutative 71.0%

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

   2. associate--l- 71.0%

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

4. Simplified 71.0%

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

5. Taylor expanded in z around 0 47.8%

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

6. Final simplification 47.8%

   \[\leadsto \log t - y \]

Alternative 9: 14.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \log t \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[Log[t], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in z around inf 38.7%

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

   1. neg-mul-1 38.7%

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

4. Simplified 38.7%

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

5. Taylor expanded in z around 0 16.0%

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

6. Final simplification 16.0%

   \[\leadsto \log t \]

Reproduce

herbie shell --seed 2023279 
(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)))