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

Percentage Accurate: 99.9% → 99.9%

Time: 9.0s

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, 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]

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.12 \cdot 10^{+18} \lor \neg \left(z \leq 3.7 \cdot 10^{+39}\right):\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log y + \log t\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.12e+18) (not (<= z 3.7e+39)))
   (- (- z) y)
   (- (+ (* x (log y)) (log t)) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.12e+18) || !(z <= 3.7e+39)) {
		tmp = -z - y;
	} else {
		tmp = ((x * log(y)) + 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 ((z <= (-1.12d+18)) .or. (.not. (z <= 3.7d+39))) then
        tmp = -z - y
    else
        tmp = ((x * log(y)) + 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 ((z <= -1.12e+18) || !(z <= 3.7e+39)) {
		tmp = -z - y;
	} else {
		tmp = ((x * Math.log(y)) + Math.log(t)) - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.12e+18) or not (z <= 3.7e+39):
		tmp = -z - y
	else:
		tmp = ((x * math.log(y)) + math.log(t)) - y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.12e+18) || !(z <= 3.7e+39))
		tmp = Float64(Float64(-z) - y);
	else
		tmp = Float64(Float64(Float64(x * log(y)) + log(t)) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.12e+18) || ~((z <= 3.7e+39)))
		tmp = -z - y;
	else
		tmp = ((x * log(y)) + log(t)) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.12e+18], N[Not[LessEqual[z, 3.7e+39]], $MachinePrecision]], N[((-z) - y), $MachinePrecision], N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[Log[t], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.12e18 or 3.70000000000000012e39 < z

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. associate--l- 100.0%

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

      3. +-commutative 100.0%

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

      4. associate--r+ 100.0%

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

      5. fma-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. remove-double-neg 100.0%

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

      10. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around inf 84.4%

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

      1. mul-1-neg 84.4%

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

   6. Simplified 84.4%

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

   if -1.12e18 < z < 3.70000000000000012e39

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. associate--l- 99.9%

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

      3. +-commutative 99.9%

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

      4. associate--r+ 99.9%

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

      5. fma-neg 99.9%

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

      6. sub-neg 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. +-commutative 99.9%

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

      9. remove-double-neg 99.9%

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

      10. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 99.2%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.12 \cdot 10^{+18} \lor \neg \left(z \leq 3.7 \cdot 10^{+39}\right):\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log y + \log t\right) - y\\ \end{array} \]

Alternative 3: 89.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+16} \lor \neg \left(x \leq 1.4 \cdot 10^{+75}\right):\\ \;\;\;\;\frac{1}{\frac{1}{x \cdot \log y}} - y\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.4e+16) (not (<= x 1.4e+75)))
   (- (/ 1.0 (/ 1.0 (* x (log y)))) y)
   (- (- (log t) z) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.4e+16) || !(x <= 1.4e+75)) {
		tmp = (1.0 / (1.0 / (x * log(y)))) - y;
	} else {
		tmp = (log(t) - z) - 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 ((x <= (-1.4d+16)) .or. (.not. (x <= 1.4d+75))) then
        tmp = (1.0d0 / (1.0d0 / (x * log(y)))) - y
    else
        tmp = (log(t) - z) - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.4e+16) || !(x <= 1.4e+75)) {
		tmp = (1.0 / (1.0 / (x * Math.log(y)))) - y;
	} else {
		tmp = (Math.log(t) - z) - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.4e+16) or not (x <= 1.4e+75):
		tmp = (1.0 / (1.0 / (x * math.log(y)))) - y
	else:
		tmp = (math.log(t) - z) - y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.4e+16) || !(x <= 1.4e+75))
		tmp = Float64(Float64(1.0 / Float64(1.0 / Float64(x * log(y)))) - y);
	else
		tmp = Float64(Float64(log(t) - z) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.4e+16) || ~((x <= 1.4e+75)))
		tmp = (1.0 / (1.0 / (x * log(y)))) - y;
	else
		tmp = (log(t) - z) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.4e+16], N[Not[LessEqual[x, 1.4e+75]], $MachinePrecision]], N[(N[(1.0 / N[(1.0 / N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4e16 or 1.40000000000000006e75 < x

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. associate--l- 99.9%

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

      3. +-commutative 99.9%

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

      4. associate--r+ 99.9%

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

      5. fma-neg 99.9%

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

      6. sub-neg 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. +-commutative 99.9%

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

      9. remove-double-neg 99.9%

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

      10. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 85.6%

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

      1. flip-+ 32.4%

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

      2. clear-num 32.3%

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

      3. clear-num 32.4%

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

      4. flip-+ 85.6%

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\log t + x \cdot \log y}}} - y \]

      5. +-commutative 85.6%

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot \log y + \log t}}} - y \]

      6. fma-udef 85.6%

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

   6. Applied egg-rr 85.6%

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

   7. Taylor expanded in x around inf 85.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x \cdot \log y}}} - y \]

   if -1.4e16 < x < 1.40000000000000006e75

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. associate--l- 100.0%

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

      3. +-commutative 100.0%

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

      4. associate--r+ 100.0%

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

      5. fma-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. remove-double-neg 100.0%

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

      10. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 96.1%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+16} \lor \neg \left(x \leq 1.4 \cdot 10^{+75}\right):\\ \;\;\;\;\frac{1}{\frac{1}{x \cdot \log y}} - y\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \end{array} \]

Alternative 4: 69.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2100000 \lor \neg \left(z \leq 2000000000\right):\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;\log t - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2100000.0) (not (<= z 2000000000.0)))
   (- (- z) y)
   (- (log t) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2100000.0) || !(z <= 2000000000.0)) {
		tmp = -z - y;
	} 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 ((z <= (-2100000.0d0)) .or. (.not. (z <= 2000000000.0d0))) then
        tmp = -z - y
    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 ((z <= -2100000.0) || !(z <= 2000000000.0)) {
		tmp = -z - y;
	} else {
		tmp = Math.log(t) - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2100000.0) or not (z <= 2000000000.0):
		tmp = -z - y
	else:
		tmp = math.log(t) - y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2100000.0) || !(z <= 2000000000.0))
		tmp = Float64(Float64(-z) - y);
	else
		tmp = Float64(log(t) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -2100000.0) || ~((z <= 2000000000.0)))
		tmp = -z - y;
	else
		tmp = log(t) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2100000.0], N[Not[LessEqual[z, 2000000000.0]], $MachinePrecision]], N[((-z) - y), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1e6 or 2e9 < z

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. associate--l- 100.0%

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

      3. +-commutative 100.0%

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

      4. associate--r+ 100.0%

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

      5. fma-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. remove-double-neg 100.0%

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

      10. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around inf 82.6%

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

      1. mul-1-neg 82.6%

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

   6. Simplified 82.6%

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

   if -2.1e6 < z < 2e9

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. associate--l- 99.9%

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

      3. +-commutative 99.9%

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

      4. associate--r+ 99.9%

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

      5. fma-neg 99.9%

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

      6. sub-neg 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. +-commutative 99.9%

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

      9. remove-double-neg 99.9%

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

      10. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 60.1%

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

   5. Taylor expanded in z around 0 59.8%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2100000 \lor \neg \left(z \leq 2000000000\right):\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;\log t - y\\ \end{array} \]

Alternative 5: 69.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 235:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 235.0) (- (log t) z) (- (- z) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 235.0) {
		tmp = log(t) - z;
	} else {
		tmp = -z - 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 <= 235.0d0) then
        tmp = log(t) - z
    else
        tmp = -z - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 235.0) {
		tmp = Math.log(t) - z;
	} else {
		tmp = -z - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 235.0:
		tmp = math.log(t) - z
	else:
		tmp = -z - y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 235.0)
		tmp = Float64(log(t) - z);
	else
		tmp = Float64(Float64(-z) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 235.0)
		tmp = log(t) - z;
	else
		tmp = -z - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 235.0], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], N[((-z) - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 235

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. associate--l- 99.9%

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

      3. +-commutative 99.9%

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

      4. associate--r+ 99.9%

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

      5. fma-neg 99.9%

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

      6. sub-neg 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. +-commutative 99.9%

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

      9. remove-double-neg 99.9%

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

      10. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 65.4%

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

   5. Taylor expanded in y around 0 65.4%

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

   if 235 < y

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. associate--l- 100.0%

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

      3. +-commutative 100.0%

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

      4. associate--r+ 100.0%

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

      5. fma-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. remove-double-neg 100.0%

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

      10. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around inf 72.9%

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

      1. mul-1-neg 72.9%

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

   6. Simplified 72.9%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 235:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \]

Alternative 6: 70.2% 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 99.9%

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

   1. associate-+l- 99.9%

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

   2. associate--l- 99.9%

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

   3. +-commutative 99.9%

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

   4. associate--r+ 99.9%

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

   5. fma-neg 99.9%

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

   6. sub-neg 99.9%

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

   7. distribute-neg-in 99.9%

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

   8. +-commutative 99.9%

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

   9. remove-double-neg 99.9%

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

   10. unsub-neg 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 70.1%

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

5. Final simplification 70.1%

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

Alternative 7: 47.5% accurate, 51.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 1.4e+77) (- z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.4e+77) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 1.4d+77) then
        tmp = -z
    else
        tmp = -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.4e+77) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 1.4e+77:
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.4e+77)
		tmp = Float64(-z);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 1.4e+77)
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.4e+77], (-z), (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 1.4e77

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. associate--l- 99.9%

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

      3. +-commutative 99.9%

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

      4. associate--r+ 99.9%

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

      5. fma-neg 99.9%

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

      6. sub-neg 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. +-commutative 99.9%

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

      9. remove-double-neg 99.9%

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

      10. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 62.9%

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

   5. Taylor expanded in z around inf 37.3%

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

      1. mul-1-neg 37.3%

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

   7. Simplified 37.3%

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

   if 1.4e77 < y

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. associate--l- 100.0%

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

      3. +-commutative 100.0%

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

      4. associate--r+ 100.0%

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

      5. fma-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. remove-double-neg 100.0%

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

      10. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 82.6%

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

   5. Taylor expanded in y around inf 72.0%

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

      1. mul-1-neg 72.0%

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

   7. Simplified 72.0%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 8: 57.1% accurate, 52.3× 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 99.9%

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

   1. associate-+l- 99.9%

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

   2. associate--l- 99.9%

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

   3. +-commutative 99.9%

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

   4. associate--r+ 99.9%

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

   5. fma-neg 99.9%

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

   6. sub-neg 99.9%

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

   7. distribute-neg-in 99.9%

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

   8. +-commutative 99.9%

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

   9. remove-double-neg 99.9%

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

   10. unsub-neg 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in z around inf 55.5%

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

   1. mul-1-neg 55.5%

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

6. Simplified 55.5%

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

7. Final simplification 55.5%

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

Alternative 9: 29.6% accurate, 104.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-+l- 99.9%

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

   2. associate--l- 99.9%

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

   3. +-commutative 99.9%

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

   4. associate--r+ 99.9%

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

   5. fma-neg 99.9%

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

   6. sub-neg 99.9%

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

   7. distribute-neg-in 99.9%

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

   8. +-commutative 99.9%

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

   9. remove-double-neg 99.9%

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

   10. unsub-neg 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 70.1%

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

5. Taylor expanded in y around inf 29.7%

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

   1. mul-1-neg 29.7%

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

7. Simplified 29.7%

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

8. Final simplification 29.7%

   \[\leadsto -y \]

Reproduce

herbie shell --seed 2023297 
(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)))