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

Percentage Accurate: 99.9% → 99.9%

Time: 6.5s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. *-rgt-identity N/A

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

   4. *-inverses N/A

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

   5. fp-cancel-sign-sub N/A

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

   6. mul-1-neg N/A

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

   7. associate-/l* N/A

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

   8. associate-*l/ N/A

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

   9. associate-*r/ N/A

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

   10. *-commutative N/A

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

   11. remove-double-neg N/A

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

   12. fp-cancel-sign-sub N/A

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

   13. mul-1-neg N/A

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

   14. *-commutative N/A

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

5. Applied rewrites 99.9%

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

Alternative 2: 68.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - y\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+18}:\\ \;\;\;\;-y\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+42}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) y)))
   (if (<= t_1 -5e+18) (- y) (if (<= t_1 5e+42) (- (log t) z) (* (log y) x)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - y;
	double tmp;
	if (t_1 <= -5e+18) {
		tmp = -y;
	} else if (t_1 <= 5e+42) {
		tmp = log(t) - z;
	} else {
		tmp = log(y) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * log(y)) - y
    if (t_1 <= (-5d+18)) then
        tmp = -y
    else if (t_1 <= 5d+42) then
        tmp = log(t) - z
    else
        tmp = log(y) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * Math.log(y)) - y;
	double tmp;
	if (t_1 <= -5e+18) {
		tmp = -y;
	} else if (t_1 <= 5e+42) {
		tmp = Math.log(t) - z;
	} else {
		tmp = Math.log(y) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - y
	tmp = 0
	if t_1 <= -5e+18:
		tmp = -y
	elif t_1 <= 5e+42:
		tmp = math.log(t) - z
	else:
		tmp = math.log(y) * x
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - y)
	tmp = 0.0
	if (t_1 <= -5e+18)
		tmp = Float64(-y);
	elseif (t_1 <= 5e+42)
		tmp = Float64(log(t) - z);
	else
		tmp = Float64(log(y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * log(y)) - y;
	tmp = 0.0;
	if (t_1 <= -5e+18)
		tmp = -y;
	elseif (t_1 <= 5e+42)
		tmp = log(t) - z;
	else
		tmp = log(y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+18], (-y), If[LessEqual[t$95$1, 5e+42], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -5e18

   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

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-rgt-identity N/A

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

      4. *-inverses N/A

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

      5. fp-cancel-sign-sub N/A

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

      6. mul-1-neg N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. remove-double-neg N/A

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

      12. fp-cancel-sign-sub N/A

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

      13. mul-1-neg N/A

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

      14. *-commutative N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 51.2

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

   8. Applied rewrites 51.2%

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

   if -5e18 < (-.f64 (*.f64 x (log.f64 y)) y) < 5.00000000000000007e42

   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 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 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.3%

      \[\leadsto \log t - z \]

   if 5.00000000000000007e42 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-rgt-identity N/A

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

      4. *-inverses N/A

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

      5. fp-cancel-sign-sub N/A

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

      6. mul-1-neg N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. remove-double-neg N/A

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

      12. fp-cancel-sign-sub N/A

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

      13. mul-1-neg N/A

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

      14. *-commutative N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 90.8

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

   8. Applied rewrites 90.8%

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

Alternative 3: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -28000000000 \lor \neg \left(x \leq 3.1 \cdot 10^{+70}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t\right) - y\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -28000000000.0) (not (<= x 3.1e+70)))
   (- (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 <= -28000000000.0) || !(x <= 3.1e+70)) {
		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 <= -28000000000.0) || !(x <= 3.1e+70))
		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, -28000000000.0], N[Not[LessEqual[x, 3.1e+70]], $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 < -2.8e10 or 3.1000000000000003e70 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 85.9

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

   5. Applied rewrites 85.9%

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

   if -2.8e10 < x < 3.1000000000000003e70

   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 97.2

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

   5. Applied rewrites 97.2%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -28000000000 \lor \neg \left(x \leq 3.1 \cdot 10^{+70}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t\right) - y\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\log y, x, \log t\right)\\ \mathbf{if}\;x \leq -28000000000:\\ \;\;\;\;t\_1 - y\\ \mathbf{elif}\;x \leq 6.1 \cdot 10^{+28}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;t\_1 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (log y) x (log t))))
   (if (<= x -28000000000.0)
     (- t_1 y)
     (if (<= x 6.1e+28) (- (- (log t) y) z) (- t_1 z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(log(y), x, log(t));
	double tmp;
	if (x <= -28000000000.0) {
		tmp = t_1 - y;
	} else if (x <= 6.1e+28) {
		tmp = (log(t) - y) - z;
	} else {
		tmp = t_1 - z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(log(y), x, log(t))
	tmp = 0.0
	if (x <= -28000000000.0)
		tmp = Float64(t_1 - y);
	elseif (x <= 6.1e+28)
		tmp = Float64(Float64(log(t) - y) - z);
	else
		tmp = Float64(t_1 - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x + N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -28000000000.0], N[(t$95$1 - y), $MachinePrecision], If[LessEqual[x, 6.1e+28], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], N[(t$95$1 - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.8e10

   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 88.7

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

   5. Applied rewrites 88.7%

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

   if -2.8e10 < x < 6.1000000000000002e28

   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 100.0

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

   5. Applied rewrites 100.0%

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

   if 6.1000000000000002e28 < x

   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 83.8

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

   5. Applied rewrites 83.8%

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

Alternative 5: 83.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.72 \cdot 10^{+139} \lor \neg \left(x \leq 3.8 \cdot 10^{+73}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.72e+139) (not (<= x 3.8e+73)))
   (* (log y) x)
   (- (- (log t) y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.72e+139) || !(x <= 3.8e+73)) {
		tmp = log(y) * x;
	} else {
		tmp = (log(t) - y) - 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 <= (-1.72d+139)) .or. (.not. (x <= 3.8d+73))) then
        tmp = log(y) * x
    else
        tmp = (log(t) - y) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.72e+139) || !(x <= 3.8e+73)) {
		tmp = Math.log(y) * x;
	} else {
		tmp = (Math.log(t) - y) - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.72e+139) or not (x <= 3.8e+73):
		tmp = math.log(y) * x
	else:
		tmp = (math.log(t) - y) - z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.72e+139) || !(x <= 3.8e+73))
		tmp = Float64(log(y) * x);
	else
		tmp = Float64(Float64(log(t) - y) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.72e+139) || ~((x <= 3.8e+73)))
		tmp = log(y) * x;
	else
		tmp = (log(t) - y) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.72e+139], N[Not[LessEqual[x, 3.8e+73]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7199999999999999e139 or 3.80000000000000022e73 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-rgt-identity N/A

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

      4. *-inverses N/A

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

      5. fp-cancel-sign-sub N/A

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

      6. mul-1-neg N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. remove-double-neg N/A

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

      12. fp-cancel-sign-sub N/A

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

      13. mul-1-neg N/A

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

      14. *-commutative N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 74.6

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

   8. Applied rewrites 74.6%

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

   if -1.7199999999999999e139 < x < 3.80000000000000022e73

   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 91.8

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

   5. Applied rewrites 91.8%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.72 \cdot 10^{+139} \lor \neg \left(x \leq 3.8 \cdot 10^{+73}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+139} \lor \neg \left(z \leq 3.8 \cdot 10^{+90}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\log t - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -8.5e+139) (not (<= z 3.8e+90))) (- z) (- (log t) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -8.5e+139) || !(z <= 3.8e+90)) {
		tmp = -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 ((z <= (-8.5d+139)) .or. (.not. (z <= 3.8d+90))) then
        tmp = -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 ((z <= -8.5e+139) || !(z <= 3.8e+90)) {
		tmp = -z;
	} else {
		tmp = Math.log(t) - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -8.5e+139) or not (z <= 3.8e+90):
		tmp = -z
	else:
		tmp = math.log(t) - y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -8.5e+139) || !(z <= 3.8e+90))
		tmp = Float64(-z);
	else
		tmp = Float64(log(t) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -8.5e+139) || ~((z <= 3.8e+90)))
		tmp = -z;
	else
		tmp = log(t) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -8.5e+139], N[Not[LessEqual[z, 3.8e+90]], $MachinePrecision]], (-z), N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.5e139 or 3.8000000000000001e90 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 72.2

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

   5. Applied rewrites 72.2%

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

   if -8.5e139 < z < 3.8000000000000001e90

   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 92.6

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.6%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+139} \lor \neg \left(z \leq 3.8 \cdot 10^{+90}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\log t - y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{+41}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 4.2e+41) (- (log t) z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.2e+41) {
		tmp = log(t) - 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 <= 4.2d+41) then
        tmp = log(t) - 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 <= 4.2e+41) {
		tmp = Math.log(t) - z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 4.2e+41:
		tmp = math.log(t) - z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 4.2e+41)
		tmp = Float64(log(t) - z);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 4.2e+41)
		tmp = log(t) - z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 4.2e+41], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 4.1999999999999999e41

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 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 95.6

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 59.7%

      \[\leadsto \log t - z \]

   if 4.1999999999999999e41 < 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

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-rgt-identity N/A

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

      4. *-inverses N/A

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

      5. fp-cancel-sign-sub N/A

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

      6. mul-1-neg N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. remove-double-neg N/A

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

      12. fp-cancel-sign-sub N/A

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

      13. mul-1-neg N/A

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

      14. *-commutative N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 59.1

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

   8. Applied rewrites 59.1%

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

Alternative 8: 47.9% accurate, 23.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{+41}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 4.2e+41) (- z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.2e+41) {
		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 <= 4.2d+41) 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 <= 4.2e+41) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 4.2e+41:
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 4.2e+41)
		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 <= 4.2e+41)
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 4.2e+41], (-z), (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 4.1999999999999999e41

   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 39.5

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

   5. Applied rewrites 39.5%

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

   if 4.1999999999999999e41 < 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

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-rgt-identity N/A

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

      4. *-inverses N/A

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

      5. fp-cancel-sign-sub N/A

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

      6. mul-1-neg N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. remove-double-neg N/A

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

      12. fp-cancel-sign-sub N/A

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

      13. mul-1-neg N/A

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

      14. *-commutative N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 59.1

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

   8. Applied rewrites 59.1%

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

Alternative 9: 30.0% accurate, 71.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. *-rgt-identity N/A

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

   4. *-inverses N/A

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

   5. fp-cancel-sign-sub N/A

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

   6. mul-1-neg N/A

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

   7. associate-/l* N/A

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

   8. associate-*l/ N/A

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

   9. associate-*r/ N/A

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

   10. *-commutative N/A

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

   11. remove-double-neg N/A

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

   12. fp-cancel-sign-sub N/A

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

   13. mul-1-neg N/A

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

   14. *-commutative N/A

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 29.7

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

8. Applied rewrites 29.7%

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

Reproduce

herbie shell --seed 2024320 
(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)))