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

Percentage Accurate: 99.9% → 99.9%

Time: 6.7s

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. Add Preprocessing
3. Add Preprocessing

Alternative 2: 56.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(x \cdot \log y - y\right) - z\right) + \log t \leq 1000:\\ \;\;\;\;\log t - y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ (- (- (* x (log y)) y) z) (log t)) 1000.0) (- (log t) y) (- z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((((x * log(y)) - y) - z) + log(t)) <= 1000.0) {
		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 (((((x * log(y)) - y) - z) + log(t)) <= 1000.0d0) 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 (((((x * Math.log(y)) - y) - z) + Math.log(t)) <= 1000.0) {
		tmp = Math.log(t) - y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((((x * math.log(y)) - y) - z) + math.log(t)) <= 1000.0:
		tmp = math.log(t) - y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * log(y)) - y) - z) + log(t)) <= 1000.0)
		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 (((((x * log(y)) - y) - z) + log(t)) <= 1000.0)
		tmp = log(t) - y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision] + N[Log[t], $MachinePrecision]), $MachinePrecision], 1000.0], N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) (log.f64 t)) < 1e3

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 82.9

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

   5. Applied rewrites 82.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 68.0%

      \[\leadsto \log t - y \]

   if 1e3 < (+.f64 (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) (log.f64 t))

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 46.2

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

   5. Applied rewrites 46.2%

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

Alternative 3: 76.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot \log y - y \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\log y}{y} \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- (* x (log y)) y) 5e+21)
   (- (- (log t) y) z)
   (* (* (/ (log y) y) x) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * log(y)) - y) <= 5e+21) {
		tmp = (log(t) - y) - z;
	} else {
		tmp = ((log(y) / y) * x) * 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 * log(y)) - y) <= 5d+21) then
        tmp = (log(t) - y) - z
    else
        tmp = ((log(y) / y) * x) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * Math.log(y)) - y) <= 5e+21) {
		tmp = (Math.log(t) - y) - z;
	} else {
		tmp = ((Math.log(y) / y) * x) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x * math.log(y)) - y) <= 5e+21:
		tmp = (math.log(t) - y) - z
	else:
		tmp = ((math.log(y) / y) * x) * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x * log(y)) - y) <= 5e+21)
		tmp = Float64(Float64(log(t) - y) - z);
	else
		tmp = Float64(Float64(Float64(log(y) / y) * x) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x * log(y)) - y) <= 5e+21)
		tmp = (log(t) - y) - z;
	else
		tmp = ((log(y) / y) * x) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision], 5e+21], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[(N[Log[y], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < 5e21

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate--r+ N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. lower--.f64 N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 86.1

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

   5. Applied rewrites 86.1%

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

   if 5e21 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 59.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 7.9%

      \[\leadsto \frac{z}{-y} \cdot y \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{x \cdot \log y}{y} \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 53.2%

       \[\leadsto \left(\frac{\log y}{y} \cdot x\right) \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+25} \lor \neg \left(x \leq 2.05 \cdot 10^{+68}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t\right) - y\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.3e+25) (not (<= x 2.05e+68)))
   (- (fma (log y) x (log t)) y)
   (- (- (log t) y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.3e+25) || !(x <= 2.05e+68)) {
		tmp = fma(log(y), x, log(t)) - y;
	} else {
		tmp = (log(t) - y) - z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.3e+25) || !(x <= 2.05e+68))
		tmp = Float64(fma(log(y), x, log(t)) - y);
	else
		tmp = Float64(Float64(log(t) - y) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.3e+25], N[Not[LessEqual[x, 2.05e+68]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] * x + N[Log[t], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2999999999999999e25 or 2.05e68 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in 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 83.9

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

   5. Applied rewrites 83.9%

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

   if -1.2999999999999999e25 < x < 2.05e68

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate--r+ N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. lower--.f64 N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 96.9

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

   5. Applied rewrites 96.9%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+25} \lor \neg \left(x \leq 2.05 \cdot 10^{+68}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t\right) - y\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.7 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 6.7e-6) (- (fma (log y) x (log t)) z) (- (- (log t) y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 6.7e-6) {
		tmp = fma(log(y), x, log(t)) - z;
	} else {
		tmp = (log(t) - y) - z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 6.7e-6)
		tmp = Float64(fma(log(y), x, log(t)) - z);
	else
		tmp = Float64(Float64(log(t) - y) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 6.7e-6], N[(N[(N[Log[y], $MachinePrecision] * x + N[Log[t], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.7e-6

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 6.7e-6 < y

   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. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. lower--.f64 N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 84.2

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

   5. Applied rewrites 84.2%

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

Alternative 6: 59.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-6}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;\log t - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.55e-6) (- (log t) z) (- (log t) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.55e-6) {
		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 <= 1.55d-6) 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 <= 1.55e-6) {
		tmp = Math.log(t) - z;
	} else {
		tmp = Math.log(t) - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 1.55e-6:
		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 <= 1.55e-6)
		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 <= 1.55e-6)
		tmp = log(t) - z;
	else
		tmp = log(t) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.55e-6], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.55e-6

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 63.5%

      \[\leadsto \log t - z \]

   if 1.55e-6 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 79.3

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

   5. Applied rewrites 79.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 63.6%

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

Alternative 7: 70.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. *-lft-identity N/A

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

   3. metadata-eval N/A

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

   4. fp-cancel-sign-sub-inv N/A

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

   5. lower--.f64 N/A

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

   6. fp-cancel-sign-sub-inv N/A

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

   7. metadata-eval N/A

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

   8. *-lft-identity N/A

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

   9. lower--.f64 N/A

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

   10. lower-log.f64 73.6

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

5. Applied rewrites 73.6%

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

Alternative 8: 48.2% accurate, 17.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{+33}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 3.5e+33) (- z) (* -1.0 y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.5e+33) {
		tmp = -z;
	} else {
		tmp = -1.0 * 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 <= 3.5d+33) then
        tmp = -z
    else
        tmp = (-1.0d0) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.5e+33) {
		tmp = -z;
	} else {
		tmp = -1.0 * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 3.5e+33:
		tmp = -z
	else:
		tmp = -1.0 * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3.5e+33)
		tmp = Float64(-z);
	else
		tmp = Float64(-1.0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 3.5e+33)
		tmp = -z;
	else
		tmp = -1.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 3.5e+33], (-z), N[(-1.0 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.5000000000000001e33

   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 32.5

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

   5. Applied rewrites 32.5%

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

   if 3.5000000000000001e33 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 67.1%

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

Alternative 9: 30.0% 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 99.9%

   \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
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 27.4

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

5. Applied rewrites 27.4%

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

Reproduce

herbie shell --seed 2024326 
(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)))