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

Percentage Accurate: 99.9% → 99.9%

Time: 10.3s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift--.f64 N/A

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

   4. associate--l- N/A

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

   5. associate-+l- N/A

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

   6. sub-neg N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. associate-+r- N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower--.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 93.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t\_1 - y\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+184}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -y\right)\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{+15}:\\ \;\;\;\;\left(\frac{t\_1}{y} - \left(\frac{z}{y} - -1\right)\right) \cdot y\\ \mathbf{elif}\;t\_2 \leq 100:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- t_1 y)))
   (if (<= t_2 -2e+184)
     (fma (log y) x (- y))
     (if (<= t_2 -5e+15)
       (* (- (/ t_1 y) (- (/ z y) -1.0)) y)
       (if (<= t_2 100.0) (- (- (log t) y) z) (fma (log y) x (- z)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = t_1 - y;
	double tmp;
	if (t_2 <= -2e+184) {
		tmp = fma(log(y), x, -y);
	} else if (t_2 <= -5e+15) {
		tmp = ((t_1 / y) - ((z / y) - -1.0)) * y;
	} else if (t_2 <= 100.0) {
		tmp = (log(t) - y) - z;
	} else {
		tmp = fma(log(y), x, -z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(t_1 - y)
	tmp = 0.0
	if (t_2 <= -2e+184)
		tmp = fma(log(y), x, Float64(-y));
	elseif (t_2 <= -5e+15)
		tmp = Float64(Float64(Float64(t_1 / y) - Float64(Float64(z / y) - -1.0)) * y);
	elseif (t_2 <= 100.0)
		tmp = Float64(Float64(log(t) - y) - z);
	else
		tmp = fma(log(y), x, Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - y), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+184], N[(N[Log[y], $MachinePrecision] * x + (-y)), $MachinePrecision], If[LessEqual[t$95$2, -5e+15], N[(N[(N[(t$95$1 / y), $MachinePrecision] - N[(N[(z / y), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$2, 100.0], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x + (-z)), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -2.00000000000000003e184

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate--l- N/A

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

      5. associate-+l- N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. associate-+r- N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 93.7

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

   7. Applied rewrites 93.7%

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

   if -2.00000000000000003e184 < (-.f64 (*.f64 x (log.f64 y)) y) < -5e15

   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 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 91.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 90.9%

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

   if -5e15 < (-.f64 (*.f64 x (log.f64 y)) y) < 100

   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. lower--.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if 100 < (-.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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate--l- N/A

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

      5. associate-+l- N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. associate-+r- N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 96.6

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

   7. Applied rewrites 96.6%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log y - y \leq -2 \cdot 10^{+184}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -y\right)\\ \mathbf{elif}\;x \cdot \log y - y \leq -5 \cdot 10^{+15}:\\ \;\;\;\;\left(\frac{x \cdot \log y}{y} - \left(\frac{z}{y} - -1\right)\right) \cdot y\\ \mathbf{elif}\;x \cdot \log y - y \leq 100:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t\_1 - y\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+278}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+22}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- t_1 y)))
   (if (<= t_2 -5e+278)
     t_1
     (if (<= t_2 -5e+22) (- (- z) y) (if (<= t_2 2e+22) (- (log t) z) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = t_1 - y;
	double tmp;
	if (t_2 <= -5e+278) {
		tmp = t_1;
	} else if (t_2 <= -5e+22) {
		tmp = -z - y;
	} else if (t_2 <= 2e+22) {
		tmp = log(t) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * log(y)
    t_2 = t_1 - y
    if (t_2 <= (-5d+278)) then
        tmp = t_1
    else if (t_2 <= (-5d+22)) then
        tmp = -z - y
    else if (t_2 <= 2d+22) then
        tmp = log(t) - z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double t_2 = t_1 - y;
	double tmp;
	if (t_2 <= -5e+278) {
		tmp = t_1;
	} else if (t_2 <= -5e+22) {
		tmp = -z - y;
	} else if (t_2 <= 2e+22) {
		tmp = Math.log(t) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = t_1 - y
	tmp = 0
	if t_2 <= -5e+278:
		tmp = t_1
	elif t_2 <= -5e+22:
		tmp = -z - y
	elif t_2 <= 2e+22:
		tmp = math.log(t) - z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(t_1 - y)
	tmp = 0.0
	if (t_2 <= -5e+278)
		tmp = t_1;
	elseif (t_2 <= -5e+22)
		tmp = Float64(Float64(-z) - y);
	elseif (t_2 <= 2e+22)
		tmp = Float64(log(t) - z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	t_2 = t_1 - y;
	tmp = 0.0;
	if (t_2 <= -5e+278)
		tmp = t_1;
	elseif (t_2 <= -5e+22)
		tmp = -z - y;
	elseif (t_2 <= 2e+22)
		tmp = log(t) - z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - y), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+278], t$95$1, If[LessEqual[t$95$2, -5e+22], N[((-z) - y), $MachinePrecision], If[LessEqual[t$95$2, 2e+22], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -5.00000000000000029e278 or 2e22 < (-.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 x around inf

      \[\leadsto \color{blue}{x \cdot \log y} \]
   4. 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 78.7

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

   5. Applied rewrites 78.7%

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

   if -5.00000000000000029e278 < (-.f64 (*.f64 x (log.f64 y)) y) < -4.9999999999999996e22

   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 inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 50.8

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

   5. Applied rewrites 50.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 74.4

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

   8. Applied rewrites 74.4%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 74.3%

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

   if -4.9999999999999996e22 < (-.f64 (*.f64 x (log.f64 y)) y) < 2e22

   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 inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 3.2

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 92.2

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

   8. Applied rewrites 92.2%

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

   9. Taylor expanded in y around 0

      \[\leadsto \log t - \color{blue}{z} \]
   10. Step-by-step derivation

   11. Applied rewrites 88.7%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log y - y \leq -5 \cdot 10^{+278}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \cdot \log y - y \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{elif}\;x \cdot \log y - y \leq 2 \cdot 10^{+22}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \log y - y\right) - z\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+22}:\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{elif}\;t\_1 \leq 10^{+34}:\\ \;\;\;\;\log t\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (- (* x (log y)) y) z)))
   (if (<= t_1 -4e+22) (- (- z) y) (if (<= t_1 1e+34) (log t) (- z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((x * log(y)) - y) - z;
	double tmp;
	if (t_1 <= -4e+22) {
		tmp = -z - y;
	} else if (t_1 <= 1e+34) {
		tmp = log(t);
	} 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) :: t_1
    real(8) :: tmp
    t_1 = ((x * log(y)) - y) - z
    if (t_1 <= (-4d+22)) then
        tmp = -z - y
    else if (t_1 <= 1d+34) then
        tmp = log(t)
    else
        tmp = -z
    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) - z;
	double tmp;
	if (t_1 <= -4e+22) {
		tmp = -z - y;
	} else if (t_1 <= 1e+34) {
		tmp = Math.log(t);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((x * math.log(y)) - y) - z
	tmp = 0
	if t_1 <= -4e+22:
		tmp = -z - y
	elif t_1 <= 1e+34:
		tmp = math.log(t)
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(x * log(y)) - y) - z)
	tmp = 0.0
	if (t_1 <= -4e+22)
		tmp = Float64(Float64(-z) - y);
	elseif (t_1 <= 1e+34)
		tmp = log(t);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((x * log(y)) - y) - z;
	tmp = 0.0;
	if (t_1 <= -4e+22)
		tmp = -z - y;
	elseif (t_1 <= 1e+34)
		tmp = log(t);
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+22], N[((-z) - y), $MachinePrecision], If[LessEqual[t$95$1, 1e+34], N[Log[t], $MachinePrecision], (-z)]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) < -4e22

   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 inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 46.7

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

   5. Applied rewrites 46.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 71.4

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

   8. Applied rewrites 71.4%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 71.4%

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

   if -4e22 < (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) < 9.99999999999999946e33

   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 inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 3.6

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

   5. Applied rewrites 3.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 81.6

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

   8. Applied rewrites 81.6%

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

   9. Taylor expanded in y around 0

      \[\leadsto \log t - \color{blue}{z} \]
   10. Step-by-step derivation

   11. Applied rewrites 75.5%

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

   12. Taylor expanded in z around 0

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

   14. Applied rewrites 72.9%

       \[\leadsto \log t \]

   if 9.99999999999999946e33 < (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z)

   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 44.8

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

   5. Applied rewrites 44.8%

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

Alternative 5: 91.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x \cdot \log y}{y} - 1, \frac{y}{z}, -1\right) \cdot z\\ \mathbf{if}\;z \leq -1.05 \cdot 10^{+113}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 12500000:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t\right) - y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma (- (/ (* x (log y)) y) 1.0) (/ y z) -1.0) z)))
   (if (<= z -1.05e+113)
     t_1
     (if (<= z 12500000.0) (- (fma (log y) x (log t)) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((((x * log(y)) / y) - 1.0), (y / z), -1.0) * z;
	double tmp;
	if (z <= -1.05e+113) {
		tmp = t_1;
	} else if (z <= 12500000.0) {
		tmp = fma(log(y), x, log(t)) - y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(Float64(Float64(Float64(x * log(y)) / y) - 1.0), Float64(y / z), -1.0) * z)
	tmp = 0.0
	if (z <= -1.05e+113)
		tmp = t_1;
	elseif (z <= 12500000.0)
		tmp = Float64(fma(log(y), x, log(t)) - y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] - 1.0), $MachinePrecision] * N[(y / z), $MachinePrecision] + -1.0), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -1.05e+113], t$95$1, If[LessEqual[z, 12500000.0], N[(N[(N[Log[y], $MachinePrecision] * x + N[Log[t], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.0499999999999999e113 or 1.25e7 < 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 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 72.4%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 93.1%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 93.1%

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

   if -1.0499999999999999e113 < z < 1.25e7

   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 96.5

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

   5. Applied rewrites 96.5%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+113}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x \cdot \log y}{y} - 1, \frac{y}{z}, -1\right) \cdot z\\ \mathbf{elif}\;z \leq 12500000:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t\right) - y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x \cdot \log y}{y} - 1, \frac{y}{z}, -1\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot \log y - y \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;\log t - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- (* x (log y)) y) -5e+22) (- (- z) y) (- (log t) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * log(y)) - y) <= -5e+22) {
		tmp = -z - y;
	} else {
		tmp = log(t) - 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) <= (-5d+22)) then
        tmp = -z - y
    else
        tmp = log(t) - 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) <= -5e+22) {
		tmp = -z - y;
	} else {
		tmp = Math.log(t) - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x * math.log(y)) - y) <= -5e+22:
		tmp = -z - y
	else:
		tmp = math.log(t) - z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x * log(y)) - y) <= -5e+22)
		tmp = Float64(Float64(-z) - y);
	else
		tmp = Float64(log(t) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x * log(y)) - y) <= -5e+22)
		tmp = -z - y;
	else
		tmp = log(t) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision], -5e+22], N[((-z) - y), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -4.9999999999999996e22

   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 inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 48.9

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

   5. Applied rewrites 48.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 70.0

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

   8. Applied rewrites 70.0%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 70.0%

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

   if -4.9999999999999996e22 < (-.f64 (*.f64 x (log.f64 y)) y)

   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 inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 2.4

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

   5. Applied rewrites 2.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 60.9

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

   8. Applied rewrites 60.9%

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

   9. Taylor expanded in y around 0

      \[\leadsto \log t - \color{blue}{z} \]
   10. Step-by-step derivation

   11. Applied rewrites 59.1%

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

Alternative 7: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t - z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot \log y}{y} - \left(\frac{z}{y} - -1\right)\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 3.3e+19)
   (fma (log y) x (- (log t) z))
   (* (- (/ (* x (log y)) y) (- (/ z y) -1.0)) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.3e+19) {
		tmp = fma(log(y), x, (log(t) - z));
	} else {
		tmp = (((x * log(y)) / y) - ((z / y) - -1.0)) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3.3e+19)
		tmp = fma(log(y), x, Float64(log(t) - z));
	else
		tmp = Float64(Float64(Float64(Float64(x * log(y)) / y) - Float64(Float64(z / y) - -1.0)) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 3.3e+19], N[(N[Log[y], $MachinePrecision] * x + N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] - N[(N[(z / y), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.3e19

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate--l- N/A

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

      5. associate-+l- N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. associate-+r- N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-log.f64 98.0

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

   7. Applied rewrites 98.0%

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

   if 3.3e19 < 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 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%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.9%

      \[\leadsto \left(\frac{\log y \cdot x}{y} - \left(\frac{z}{y} - -1\right)\right) \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t - z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot \log y}{y} - \left(\frac{z}{y} - -1\right)\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot \log y}{y} - \left(\frac{z}{y} - -1\right)\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 3.3e+19)
   (- (fma (log y) x (log t)) z)
   (* (- (/ (* x (log y)) y) (- (/ z y) -1.0)) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.3e+19) {
		tmp = fma(log(y), x, log(t)) - z;
	} else {
		tmp = (((x * log(y)) / y) - ((z / y) - -1.0)) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3.3e+19)
		tmp = Float64(fma(log(y), x, log(t)) - z);
	else
		tmp = Float64(Float64(Float64(Float64(x * log(y)) / y) - Float64(Float64(z / y) - -1.0)) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 3.3e+19], N[(N[(N[Log[y], $MachinePrecision] * x + N[Log[t], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] - N[(N[(z / y), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.3e19

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 97.9

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

   5. Applied rewrites 97.9%

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

   if 3.3e19 < 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 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%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.9%

      \[\leadsto \left(\frac{\log y \cdot x}{y} - \left(\frac{z}{y} - -1\right)\right) \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot \log y}{y} - \left(\frac{z}{y} - -1\right)\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 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 99.8%

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

Alternative 10: 89.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -y\right)\\ \mathbf{elif}\;x \leq 30000000000000:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.4e+20)
   (fma (log y) x (- y))
   (if (<= x 30000000000000.0) (- (- (log t) y) z) (fma (log y) x (- z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.4e+20) {
		tmp = fma(log(y), x, -y);
	} else if (x <= 30000000000000.0) {
		tmp = (log(t) - y) - z;
	} else {
		tmp = fma(log(y), x, -z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.4e+20)
		tmp = fma(log(y), x, Float64(-y));
	elseif (x <= 30000000000000.0)
		tmp = Float64(Float64(log(t) - y) - z);
	else
		tmp = fma(log(y), x, Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.4e+20], N[(N[Log[y], $MachinePrecision] * x + (-y)), $MachinePrecision], If[LessEqual[x, 30000000000000.0], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x + (-z)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.4e20

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate--l- N/A

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

      5. associate-+l- N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. associate-+r- N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 87.9

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

   7. Applied rewrites 87.9%

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

   if -2.4e20 < x < 3e13

   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. lower--.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 97.2

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

   5. Applied rewrites 97.2%

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

   if 3e13 < x

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate--l- N/A

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

      5. associate-+l- N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. associate-+r- N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 83.9

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

   7. Applied rewrites 83.9%

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

Alternative 11: 89.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\log y, x, -y\right)\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+77}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (log y) x (- y))))
   (if (<= x -2.4e+20) t_1 (if (<= x 1.12e+77) (- (- (log t) y) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(log(y), x, -y);
	double tmp;
	if (x <= -2.4e+20) {
		tmp = t_1;
	} else if (x <= 1.12e+77) {
		tmp = (log(t) - y) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(log(y), x, Float64(-y))
	tmp = 0.0
	if (x <= -2.4e+20)
		tmp = t_1;
	elseif (x <= 1.12e+77)
		tmp = Float64(Float64(log(t) - y) - z);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x + (-y)), $MachinePrecision]}, If[LessEqual[x, -2.4e+20], t$95$1, If[LessEqual[x, 1.12e+77], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4e20 or 1.1199999999999999e77 < x

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate--l- N/A

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

      5. associate-+l- N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. associate-+r- N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 87.8

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

   7. Applied rewrites 87.8%

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

   if -2.4e20 < x < 1.1199999999999999e77

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 93.2

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

   5. Applied rewrites 93.2%

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

Alternative 12: 85.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -9.5 \cdot 10^{+99}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{+144}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -9.5e+99) t_1 (if (<= x 2.65e+144) (- (- (log t) y) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -9.5e+99) {
		tmp = t_1;
	} else if (x <= 2.65e+144) {
		tmp = (log(t) - y) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if (x <= (-9.5d+99)) then
        tmp = t_1
    else if (x <= 2.65d+144) then
        tmp = (log(t) - y) - z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double tmp;
	if (x <= -9.5e+99) {
		tmp = t_1;
	} else if (x <= 2.65e+144) {
		tmp = (Math.log(t) - y) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -9.5e+99:
		tmp = t_1
	elif x <= 2.65e+144:
		tmp = (math.log(t) - y) - z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -9.5e+99)
		tmp = t_1;
	elseif (x <= 2.65e+144)
		tmp = Float64(Float64(log(t) - y) - z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (x <= -9.5e+99)
		tmp = t_1;
	elseif (x <= 2.65e+144)
		tmp = (log(t) - y) - z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9.5e+99], t$95$1, If[LessEqual[x, 2.65e+144], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.49999999999999908e99 or 2.6499999999999998e144 < 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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 79.7

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

   5. Applied rewrites 79.7%

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

   if -9.49999999999999908e99 < x < 2.6499999999999998e144

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 88.6

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

   5. Applied rewrites 88.6%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+99}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{+144}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 47.8% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+112}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 21.5:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -8e+112) (- z) (if (<= z 21.5) (- y) (- z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -8e+112) {
		tmp = -z;
	} else if (z <= 21.5) {
		tmp = -y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-8d+112)) then
        tmp = -z
    else if (z <= 21.5d0) then
        tmp = -y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -8e+112) {
		tmp = -z;
	} else if (z <= 21.5) {
		tmp = -y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -8e+112:
		tmp = -z
	elif z <= 21.5:
		tmp = -y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -8e+112)
		tmp = Float64(-z);
	elseif (z <= 21.5)
		tmp = Float64(-y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -8e+112)
		tmp = -z;
	elseif (z <= 21.5)
		tmp = -y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -8e+112], (-z), If[LessEqual[z, 21.5], (-y), (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -7.9999999999999994e112 or 21.5 < 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 63.0

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

   5. Applied rewrites 63.0%

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

   if -7.9999999999999994e112 < z < 21.5

   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 inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 36.8

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

   5. Applied rewrites 36.8%

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

Alternative 14: 58.4% accurate, 35.8× 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 99.8%

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

3. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 27.8

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

5. Applied rewrites 27.8%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate--r+ N/A

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

   3. lower--.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-log.f64 65.9

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

8. Applied rewrites 65.9%

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

9. Taylor expanded in z around inf

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

11. Applied rewrites 53.6%

    \[\leadsto \left(-z\right) - y \]
12. Add Preprocessing

Alternative 15: 30.5% 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 99.8%

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

3. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 27.8

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

5. Applied rewrites 27.8%

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

Reproduce

herbie shell --seed 2024277 
(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)))