Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2

Percentage Accurate: 89.2% → 99.6%

Time: 17.0s

Alternatives: 18

Speedup: 1.9×

Specification

Math

\[\begin{array}{l} \\ \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))

C

double code(double x, double y, double z, double t) {
	return (((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)))) - 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 - 1.0d0) * log(y)) + ((z - 1.0d0) * log((1.0d0 - y)))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((x - 1.0) * Math.log(y)) + ((z - 1.0) * Math.log((1.0 - y)))) - t;
}

Python

def code(x, y, z, t):
	return (((x - 1.0) * math.log(y)) + ((z - 1.0) * math.log((1.0 - y)))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * log(Float64(1.0 - y)))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)))) - t;
end

Wolfram

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

Initial Program: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))

C

double code(double x, double y, double z, double t) {
	return (((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)))) - 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 - 1.0d0) * log(y)) + ((z - 1.0d0) * log((1.0d0 - y)))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((x - 1.0) * Math.log(y)) + ((z - 1.0) * Math.log((1.0 - y)))) - t;
}

Python

def code(x, y, z, t):
	return (((x - 1.0) * math.log(y)) + ((z - 1.0) * math.log((1.0 - y)))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * log(Float64(1.0 - y)))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)))) - t;
end

Wolfram

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

Alternative 1: 99.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(z + -1\right) \cdot \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) + \left(x + -1\right) \cdot \log y\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (+
   (*
    (+ z -1.0)
    (* y (fma y (fma y (fma y -0.25 -0.3333333333333333) -0.5) -1.0)))
   (* (+ x -1.0) (log y)))
  t))

C

double code(double x, double y, double z, double t) {
	return (((z + -1.0) * (y * fma(y, fma(y, fma(y, -0.25, -0.3333333333333333), -0.5), -1.0))) + ((x + -1.0) * log(y))) - t;
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(z + -1.0) * Float64(y * fma(y, fma(y, fma(y, -0.25, -0.3333333333333333), -0.5), -1.0))) + Float64(Float64(x + -1.0) * log(y))) - t)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(z + -1.0), $MachinePrecision] * N[(y * N[(y * N[(y * N[(y * -0.25 + -0.3333333333333333), $MachinePrecision] + -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x + -1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 91.7%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}\right) - t \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}\right) - t \]

   2. sub-neg N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) - t \]

   3. metadata-eval N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)\right)\right) - t \]

   4. lower-fma.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}\right)\right) - t \]

   5. sub-neg N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right)\right) - t \]

   6. metadata-eval N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)\right)\right) - t \]

   7. lower-fma.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(y \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{-1}{4} \cdot y - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)\right)\right) - t \]

   8. sub-neg N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\frac{-1}{4} \cdot y + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)\right)\right) - t \]

   9. *-commutative N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)\right)\right) - t \]

   10. metadata-eval N/A

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, y \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)\right)\right) - t \]

   11. lower-fma.f64 99.9

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)\right)\right) - t \]

5. Applied rewrites 99.9%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right)}\right) - t \]

6. Final simplification 99.9%

   \[\leadsto \left(\left(z + -1\right) \cdot \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) + \left(x + -1\right) \cdot \log y\right) - t \]
7. Add Preprocessing

Alternative 2: 91.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(x + -1\right) \cdot \log y + \left(z + -1\right) \cdot \log \left(1 - y\right)\right) - t\\ t_2 := x \cdot \log y - t\\ \mathbf{if}\;t\_1 \leq -20000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1000:\\ \;\;\;\;\left(-\log y\right) - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (+ (* (+ x -1.0) (log y)) (* (+ z -1.0) (log (- 1.0 y)))) t))
        (t_2 (- (* x (log y)) t)))
   (if (<= t_1 -20000000000.0)
     t_2
     (if (<= t_1 1000.0) (- (- (log y)) (* y z)) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (((x + -1.0) * log(y)) + ((z + -1.0) * log((1.0 - y)))) - t;
	double t_2 = (x * log(y)) - t;
	double tmp;
	if (t_1 <= -20000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 1000.0) {
		tmp = -log(y) - (y * z);
	} else {
		tmp = t_2;
	}
	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 + (-1.0d0)) * log(y)) + ((z + (-1.0d0)) * log((1.0d0 - y)))) - t
    t_2 = (x * log(y)) - t
    if (t_1 <= (-20000000000.0d0)) then
        tmp = t_2
    else if (t_1 <= 1000.0d0) then
        tmp = -log(y) - (y * z)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (((x + -1.0) * Math.log(y)) + ((z + -1.0) * Math.log((1.0 - y)))) - t;
	double t_2 = (x * Math.log(y)) - t;
	double tmp;
	if (t_1 <= -20000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 1000.0) {
		tmp = -Math.log(y) - (y * z);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (((x + -1.0) * math.log(y)) + ((z + -1.0) * math.log((1.0 - y)))) - t
	t_2 = (x * math.log(y)) - t
	tmp = 0
	if t_1 <= -20000000000.0:
		tmp = t_2
	elif t_1 <= 1000.0:
		tmp = -math.log(y) - (y * z)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(x + -1.0) * log(y)) + Float64(Float64(z + -1.0) * log(Float64(1.0 - y)))) - t)
	t_2 = Float64(Float64(x * log(y)) - t)
	tmp = 0.0
	if (t_1 <= -20000000000.0)
		tmp = t_2;
	elseif (t_1 <= 1000.0)
		tmp = Float64(Float64(-log(y)) - Float64(y * z));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (((x + -1.0) * log(y)) + ((z + -1.0) * log((1.0 - y)))) - t;
	t_2 = (x * log(y)) - t;
	tmp = 0.0;
	if (t_1 <= -20000000000.0)
		tmp = t_2;
	elseif (t_1 <= 1000.0)
		tmp = -log(y) - (y * z);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(x + -1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[(z + -1.0), $MachinePrecision] * N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -20000000000.0], t$95$2, If[LessEqual[t$95$1, 1000.0], N[((-N[Log[y], $MachinePrecision]) - N[(y * z), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) t) < -2e10 or 1e3 < (-.f64 (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) t)

   1. Initial program 94.6%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 93.4

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

   5. Applied rewrites 93.4%

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

   if -2e10 < (-.f64 (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) t) < 1e3

   1. Initial program 82.4%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate--l- N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.0%

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

   9. Taylor expanded in z around inf

      \[\leadsto \left(\mathsf{neg}\left(\log y\right)\right) - y \cdot \color{blue}{z} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.6%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(x + -1\right) \cdot \log y + \left(z + -1\right) \cdot \log \left(1 - y\right)\right) - t \leq -20000000000:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;\left(\left(x + -1\right) \cdot \log y + \left(z + -1\right) \cdot \log \left(1 - y\right)\right) - t \leq 1000:\\ \;\;\;\;\left(-\log y\right) - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + -1\right) \cdot \log y - t\\ \mathbf{if}\;x + -1 \leq -1.0000000005:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x + -1 \leq -1:\\ \;\;\;\;\left(-\log y\right) - \mathsf{fma}\left(y, z + -1, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (+ x -1.0) (log y)) t)))
   (if (<= (+ x -1.0) -1.0000000005)
     t_1
     (if (<= (+ x -1.0) -1.0) (- (- (log y)) (fma y (+ z -1.0) t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((x + -1.0) * log(y)) - t;
	double tmp;
	if ((x + -1.0) <= -1.0000000005) {
		tmp = t_1;
	} else if ((x + -1.0) <= -1.0) {
		tmp = -log(y) - fma(y, (z + -1.0), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(x + -1.0) * log(y)) - t)
	tmp = 0.0
	if (Float64(x + -1.0) <= -1.0000000005)
		tmp = t_1;
	elseif (Float64(x + -1.0) <= -1.0)
		tmp = Float64(Float64(-log(y)) - fma(y, Float64(z + -1.0), t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(x + -1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[N[(x + -1.0), $MachinePrecision], -1.0000000005], t$95$1, If[LessEqual[N[(x + -1.0), $MachinePrecision], -1.0], N[((-N[Log[y], $MachinePrecision]) - N[(y * N[(z + -1.0), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -1.0000000005 or -1 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 94.5%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 93.9

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

   5. Applied rewrites 93.9%

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

   if -1.0000000005 < (-.f64 x #s(literal 1 binary64)) < -1

   1. Initial program 88.5%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate--l- N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

      \[\leadsto \left(-\log y\right) - \mathsf{fma}\left(\color{blue}{y}, -1 + z, t\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + -1 \leq -1.0000000005:\\ \;\;\;\;\left(x + -1\right) \cdot \log y - t\\ \mathbf{elif}\;x + -1 \leq -1:\\ \;\;\;\;\left(-\log y\right) - \mathsf{fma}\left(y, z + -1, t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + -1\right) \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + -1\right) \cdot \log y\\ t_2 := t\_1 - t\\ \mathbf{if}\;t \leq -5.2:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - z, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (+ x -1.0) (log y))) (t_2 (- t_1 t)))
   (if (<= t -5.2) t_2 (if (<= t 1.1e-29) (fma y (- 1.0 z) t_1) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + -1.0) * log(y);
	double t_2 = t_1 - t;
	double tmp;
	if (t <= -5.2) {
		tmp = t_2;
	} else if (t <= 1.1e-29) {
		tmp = fma(y, (1.0 - z), t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + -1.0) * log(y))
	t_2 = Float64(t_1 - t)
	tmp = 0.0
	if (t <= -5.2)
		tmp = t_2;
	elseif (t <= 1.1e-29)
		tmp = fma(y, Float64(1.0 - z), t_1);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + -1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - t), $MachinePrecision]}, If[LessEqual[t, -5.2], t$95$2, If[LessEqual[t, 1.1e-29], N[(y * N[(1.0 - z), $MachinePrecision] + t$95$1), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.20000000000000018 or 1.09999999999999995e-29 < t

   1. Initial program 96.1%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 96.1

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

   5. Applied rewrites 96.1%

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

   if -5.20000000000000018 < t < 1.09999999999999995e-29

   1. Initial program 86.9%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate--l- N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 98.9%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{1 - z}, \log y \cdot \left(-1 + x\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.2:\\ \;\;\;\;\left(x + -1\right) \cdot \log y - t\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - z, \left(x + -1\right) \cdot \log y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + -1\right) \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + -1\right) \cdot \log y\\ t_2 := t\_1 - t\\ \mathbf{if}\;t \leq -5.2:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-29}:\\ \;\;\;\;t\_1 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (+ x -1.0) (log y))) (t_2 (- t_1 t)))
   (if (<= t -5.2) t_2 (if (<= t 1.1e-29) (- t_1 (* y z)) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + -1.0) * log(y);
	double t_2 = t_1 - t;
	double tmp;
	if (t <= -5.2) {
		tmp = t_2;
	} else if (t <= 1.1e-29) {
		tmp = t_1 - (y * z);
	} else {
		tmp = t_2;
	}
	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 + (-1.0d0)) * log(y)
    t_2 = t_1 - t
    if (t <= (-5.2d0)) then
        tmp = t_2
    else if (t <= 1.1d-29) then
        tmp = t_1 - (y * z)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + -1.0) * Math.log(y);
	double t_2 = t_1 - t;
	double tmp;
	if (t <= -5.2) {
		tmp = t_2;
	} else if (t <= 1.1e-29) {
		tmp = t_1 - (y * z);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + -1.0) * math.log(y)
	t_2 = t_1 - t
	tmp = 0
	if t <= -5.2:
		tmp = t_2
	elif t <= 1.1e-29:
		tmp = t_1 - (y * z)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + -1.0) * log(y))
	t_2 = Float64(t_1 - t)
	tmp = 0.0
	if (t <= -5.2)
		tmp = t_2;
	elseif (t <= 1.1e-29)
		tmp = Float64(t_1 - Float64(y * z));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + -1.0) * log(y);
	t_2 = t_1 - t;
	tmp = 0.0;
	if (t <= -5.2)
		tmp = t_2;
	elseif (t <= 1.1e-29)
		tmp = t_1 - (y * z);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + -1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - t), $MachinePrecision]}, If[LessEqual[t, -5.2], t$95$2, If[LessEqual[t, 1.1e-29], N[(t$95$1 - N[(y * z), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.20000000000000018 or 1.09999999999999995e-29 < t

   1. Initial program 96.1%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 96.1

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

   5. Applied rewrites 96.1%

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

   if -5.20000000000000018 < t < 1.09999999999999995e-29

   1. Initial program 86.9%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate--l- N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 98.8%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.2:\\ \;\;\;\;\left(x + -1\right) \cdot \log y - t\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-29}:\\ \;\;\;\;\left(x + -1\right) \cdot \log y - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(x + -1\right) \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;x + -1 \leq -1000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x + -1 \leq -0.5:\\ \;\;\;\;\left(-\log y\right) - \left(t - y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t)))
   (if (<= (+ x -1.0) -1000.0)
     t_1
     (if (<= (+ x -1.0) -0.5) (- (- (log y)) (- t y)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - t;
	double tmp;
	if ((x + -1.0) <= -1000.0) {
		tmp = t_1;
	} else if ((x + -1.0) <= -0.5) {
		tmp = -log(y) - (t - y);
	} 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)) - t
    if ((x + (-1.0d0)) <= (-1000.0d0)) then
        tmp = t_1
    else if ((x + (-1.0d0)) <= (-0.5d0)) then
        tmp = -log(y) - (t - y)
    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)) - t;
	double tmp;
	if ((x + -1.0) <= -1000.0) {
		tmp = t_1;
	} else if ((x + -1.0) <= -0.5) {
		tmp = -Math.log(y) - (t - y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - t
	tmp = 0
	if (x + -1.0) <= -1000.0:
		tmp = t_1
	elif (x + -1.0) <= -0.5:
		tmp = -math.log(y) - (t - y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - t)
	tmp = 0.0
	if (Float64(x + -1.0) <= -1000.0)
		tmp = t_1;
	elseif (Float64(x + -1.0) <= -0.5)
		tmp = Float64(Float64(-log(y)) - Float64(t - y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * log(y)) - t;
	tmp = 0.0;
	if ((x + -1.0) <= -1000.0)
		tmp = t_1;
	elseif ((x + -1.0) <= -0.5)
		tmp = -log(y) - (t - y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[N[(x + -1.0), $MachinePrecision], -1000.0], t$95$1, If[LessEqual[N[(x + -1.0), $MachinePrecision], -0.5], N[((-N[Log[y], $MachinePrecision]) - N[(t - y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -1e3 or -0.5 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 94.3%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 92.9

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

   5. Applied rewrites 92.9%

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

   if -1e3 < (-.f64 x #s(literal 1 binary64)) < -0.5

   1. Initial program 89.0%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate--l- N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.0%

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

   9. Taylor expanded in z around 0

      \[\leadsto \left(\mathsf{neg}\left(\log y\right)\right) - \left(t + \color{blue}{-1 \cdot y}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 88.1%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + -1 \leq -1000:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x + -1 \leq -0.5:\\ \;\;\;\;\left(-\log y\right) - \left(t - y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x + -1 \leq -1 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x + -1 \leq -0.5:\\ \;\;\;\;\left(-\log y\right) - \left(t - y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= (+ x -1.0) -1e+80)
     t_1
     (if (<= (+ x -1.0) -0.5) (- (- (log y)) (- t y)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if ((x + -1.0) <= -1e+80) {
		tmp = t_1;
	} else if ((x + -1.0) <= -0.5) {
		tmp = -log(y) - (t - y);
	} 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 + (-1.0d0)) <= (-1d+80)) then
        tmp = t_1
    else if ((x + (-1.0d0)) <= (-0.5d0)) then
        tmp = -log(y) - (t - y)
    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 + -1.0) <= -1e+80) {
		tmp = t_1;
	} else if ((x + -1.0) <= -0.5) {
		tmp = -Math.log(y) - (t - y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if (x + -1.0) <= -1e+80:
		tmp = t_1
	elif (x + -1.0) <= -0.5:
		tmp = -math.log(y) - (t - y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (Float64(x + -1.0) <= -1e+80)
		tmp = t_1;
	elseif (Float64(x + -1.0) <= -0.5)
		tmp = Float64(Float64(-log(y)) - Float64(t - y));
	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 + -1.0) <= -1e+80)
		tmp = t_1;
	elseif ((x + -1.0) <= -0.5)
		tmp = -log(y) - (t - y);
	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[N[(x + -1.0), $MachinePrecision], -1e+80], t$95$1, If[LessEqual[N[(x + -1.0), $MachinePrecision], -0.5], N[((-N[Log[y], $MachinePrecision]) - N[(t - y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -1e80 or -0.5 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 95.8%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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 74.5

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

   5. Applied rewrites 74.5%

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

   if -1e80 < (-.f64 x #s(literal 1 binary64)) < -0.5

   1. Initial program 88.6%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate--l- N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.2%

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

   9. Taylor expanded in z around 0

      \[\leadsto \left(\mathsf{neg}\left(\log y\right)\right) - \left(t + \color{blue}{-1 \cdot y}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 83.0%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + -1 \leq -1 \cdot 10^{+80}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x + -1 \leq -0.5:\\ \;\;\;\;\left(-\log y\right) - \left(t - y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x + -1 \leq -1 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x + -1 \leq 5 \cdot 10^{+57}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= (+ x -1.0) -1e+80)
     t_1
     (if (<= (+ x -1.0) 5e+57) (- (* z (- y)) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if ((x + -1.0) <= -1e+80) {
		tmp = t_1;
	} else if ((x + -1.0) <= 5e+57) {
		tmp = (z * -y) - t;
	} 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 + (-1.0d0)) <= (-1d+80)) then
        tmp = t_1
    else if ((x + (-1.0d0)) <= 5d+57) then
        tmp = (z * -y) - t
    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 + -1.0) <= -1e+80) {
		tmp = t_1;
	} else if ((x + -1.0) <= 5e+57) {
		tmp = (z * -y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if (x + -1.0) <= -1e+80:
		tmp = t_1
	elif (x + -1.0) <= 5e+57:
		tmp = (z * -y) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (Float64(x + -1.0) <= -1e+80)
		tmp = t_1;
	elseif (Float64(x + -1.0) <= 5e+57)
		tmp = Float64(Float64(z * Float64(-y)) - t);
	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 + -1.0) <= -1e+80)
		tmp = t_1;
	elseif ((x + -1.0) <= 5e+57)
		tmp = (z * -y) - t;
	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[N[(x + -1.0), $MachinePrecision], -1e+80], t$95$1, If[LessEqual[N[(x + -1.0), $MachinePrecision], 5e+57], N[(N[(z * (-y)), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -1e80 or 4.99999999999999972e57 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 97.2%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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 79.9

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

   5. Applied rewrites 79.9%

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

   if -1e80 < (-.f64 x #s(literal 1 binary64)) < 4.99999999999999972e57

   1. Initial program 88.5%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

         \[\leadsto \left(\log y \cdot \color{blue}{\frac{{x}^{3} - {1}^{3}}{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

      5. clear-num N/A

         \[\leadsto \left(\log y \cdot \color{blue}{\frac{1}{\frac{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}{{x}^{3} - {1}^{3}}}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

      6. un-div-inv N/A

         \[\leadsto \left(\color{blue}{\frac{\log y}{\frac{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}{{x}^{3} - {1}^{3}}}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\log y}{\frac{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}{{x}^{3} - {1}^{3}}}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

      8. clear-num N/A

         \[\leadsto \left(\frac{\log y}{\color{blue}{\frac{1}{\frac{{x}^{3} - {1}^{3}}{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}}}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

      9. flip3-- N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 88.5

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. lower-+.f64 N/A

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

      15. metadata-eval 88.5

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

   4. Applied rewrites 88.5%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-log1p.f64 N/A

         \[\leadsto \color{blue}{\mathsf{log1p}\left(-1 \cdot y\right)} \cdot z - t \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{neg}\left(y\right)}\right) \cdot z - t \]

      7. lower-neg.f64 60.3

         \[\leadsto \mathsf{log1p}\left(\color{blue}{-y}\right) \cdot z - t \]

   7. Applied rewrites 60.3%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(-y\right) \cdot z} - t \]

   8. Taylor expanded in y around 0

      \[\leadsto \left(-1 \cdot y\right) \cdot z - t \]
   9. Step-by-step derivation

   10. Applied rewrites 60.3%

       \[\leadsto \left(-y\right) \cdot z - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + -1 \leq -1 \cdot 10^{+80}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x + -1 \leq 5 \cdot 10^{+57}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 88.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z + -1 \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\left(x + -1\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ z -1.0) 5e+255) (- (* (+ x -1.0) (log y)) t) (- (* z (- y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z + -1.0) <= 5e+255) {
		tmp = ((x + -1.0) * log(y)) - t;
	} else {
		tmp = (z * -y) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z + (-1.0d0)) <= 5d+255) then
        tmp = ((x + (-1.0d0)) * log(y)) - t
    else
        tmp = (z * -y) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z + -1.0) <= 5e+255) {
		tmp = ((x + -1.0) * Math.log(y)) - t;
	} else {
		tmp = (z * -y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z + -1.0) <= 5e+255:
		tmp = ((x + -1.0) * math.log(y)) - t
	else:
		tmp = (z * -y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z + -1.0) <= 5e+255)
		tmp = Float64(Float64(Float64(x + -1.0) * log(y)) - t);
	else
		tmp = Float64(Float64(z * Float64(-y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z + -1.0) <= 5e+255)
		tmp = ((x + -1.0) * log(y)) - t;
	else
		tmp = (z * -y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z + -1.0), $MachinePrecision], 5e+255], N[(N[(N[(x + -1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(z * (-y)), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z #s(literal 1 binary64)) < 5.0000000000000002e255

   1. Initial program 93.8%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 93.5

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

   5. Applied rewrites 93.5%

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

   if 5.0000000000000002e255 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 53.3%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

         \[\leadsto \left(\log y \cdot \color{blue}{\frac{{x}^{3} - {1}^{3}}{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

      5. clear-num N/A

         \[\leadsto \left(\log y \cdot \color{blue}{\frac{1}{\frac{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}{{x}^{3} - {1}^{3}}}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

      6. un-div-inv N/A

         \[\leadsto \left(\color{blue}{\frac{\log y}{\frac{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}{{x}^{3} - {1}^{3}}}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\log y}{\frac{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}{{x}^{3} - {1}^{3}}}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

      8. clear-num N/A

         \[\leadsto \left(\frac{\log y}{\color{blue}{\frac{1}{\frac{{x}^{3} - {1}^{3}}{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}}}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

      9. flip3-- N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 53.3

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. lower-+.f64 N/A

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

      15. metadata-eval 53.3

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

   4. Applied rewrites 53.3%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-log1p.f64 N/A

         \[\leadsto \color{blue}{\mathsf{log1p}\left(-1 \cdot y\right)} \cdot z - t \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{neg}\left(y\right)}\right) \cdot z - t \]

      7. lower-neg.f64 99.9

         \[\leadsto \mathsf{log1p}\left(\color{blue}{-y}\right) \cdot z - t \]

   7. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(-y\right) \cdot z} - t \]

   8. Taylor expanded in y around 0

      \[\leadsto \left(-1 \cdot y\right) \cdot z - t \]
   9. Step-by-step derivation

   10. Applied rewrites 99.9%

       \[\leadsto \left(-y\right) \cdot z - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + -1 \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\left(x + -1\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(x + -1\right) \cdot \log y - \mathsf{fma}\left(y, z + -1, t\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (* (+ x -1.0) (log y)) (fma y (+ z -1.0) t)))

C

double code(double x, double y, double z, double t) {
	return ((x + -1.0) * log(y)) - fma(y, (z + -1.0), t);
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x + -1.0) * log(y)) - fma(y, Float64(z + -1.0), t))
end

Wolfram

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

Derivation

1. Initial program 91.7%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. associate--l- N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-log.f64 N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. sub-neg N/A

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

   14. metadata-eval N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 99.7

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

5. Applied rewrites 99.7%

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

6. Final simplification 99.7%

   \[\leadsto \left(x + -1\right) \cdot \log y - \mathsf{fma}\left(y, z + -1, t\right) \]
7. Add Preprocessing

Alternative 11: 46.4% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ z \cdot \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (* z (* y (fma y (fma y (fma y -0.25 -0.3333333333333333) -0.5) -1.0))) t))

C

double code(double x, double y, double z, double t) {
	return (z * (y * fma(y, fma(y, fma(y, -0.25, -0.3333333333333333), -0.5), -1.0))) - t;
}

Julia

function code(x, y, z, t)
	return Float64(Float64(z * Float64(y * fma(y, fma(y, fma(y, -0.25, -0.3333333333333333), -0.5), -1.0))) - t)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(z * N[(y * N[(y * N[(y * N[(y * -0.25 + -0.3333333333333333), $MachinePrecision] + -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 91.7%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. flip3-- N/A

      \[\leadsto \left(\log y \cdot \color{blue}{\frac{{x}^{3} - {1}^{3}}{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   5. clear-num N/A

      \[\leadsto \left(\log y \cdot \color{blue}{\frac{1}{\frac{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}{{x}^{3} - {1}^{3}}}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   6. un-div-inv N/A

      \[\leadsto \left(\color{blue}{\frac{\log y}{\frac{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}{{x}^{3} - {1}^{3}}}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   7. lower-/.f64 N/A

      \[\leadsto \left(\color{blue}{\frac{\log y}{\frac{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}{{x}^{3} - {1}^{3}}}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   8. clear-num N/A

      \[\leadsto \left(\frac{\log y}{\color{blue}{\frac{1}{\frac{{x}^{3} - {1}^{3}}{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}}}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   9. flip3-- N/A

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

   10. lift--.f64 N/A

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

   11. lower-/.f64 91.6

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

   12. lift--.f64 N/A

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

   13. sub-neg N/A

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

   14. lower-+.f64 N/A

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

   15. metadata-eval 91.6

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

4. Applied rewrites 91.6%

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

5. Taylor expanded in z around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. mul-1-neg N/A

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

   5. lower-log1p.f64 N/A

      \[\leadsto \color{blue}{\mathsf{log1p}\left(-1 \cdot y\right)} \cdot z - t \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{neg}\left(y\right)}\right) \cdot z - t \]

   7. lower-neg.f64 45.6

      \[\leadsto \mathsf{log1p}\left(\color{blue}{-y}\right) \cdot z - t \]

7. Applied rewrites 45.6%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(-y\right) \cdot z} - t \]

8. Taylor expanded in y around 0

   \[\leadsto \left(y \cdot \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right) \cdot z - t \]
9. Step-by-step derivation

10. Applied rewrites 45.7%

    \[\leadsto \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) \cdot z - t \]

11. Final simplification 45.7%

    \[\leadsto z \cdot \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) - t \]
12. Add Preprocessing

Alternative 12: 46.3% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ y \cdot \mathsf{fma}\left(y, z \cdot \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), -z\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (* y (fma y (* z (fma y -0.3333333333333333 -0.5)) (- z))) t))

C

double code(double x, double y, double z, double t) {
	return (y * fma(y, (z * fma(y, -0.3333333333333333, -0.5)), -z)) - t;
}

Julia

function code(x, y, z, t)
	return Float64(Float64(y * fma(y, Float64(z * fma(y, -0.3333333333333333, -0.5)), Float64(-z))) - t)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(y * N[(y * N[(z * N[(y * -0.3333333333333333 + -0.5), $MachinePrecision]), $MachinePrecision] + (-z)), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 91.7%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. flip3-- N/A

      \[\leadsto \left(\log y \cdot \color{blue}{\frac{{x}^{3} - {1}^{3}}{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   5. clear-num N/A

      \[\leadsto \left(\log y \cdot \color{blue}{\frac{1}{\frac{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}{{x}^{3} - {1}^{3}}}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   6. un-div-inv N/A

      \[\leadsto \left(\color{blue}{\frac{\log y}{\frac{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}{{x}^{3} - {1}^{3}}}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   7. lower-/.f64 N/A

      \[\leadsto \left(\color{blue}{\frac{\log y}{\frac{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}{{x}^{3} - {1}^{3}}}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   8. clear-num N/A

      \[\leadsto \left(\frac{\log y}{\color{blue}{\frac{1}{\frac{{x}^{3} - {1}^{3}}{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}}}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   9. flip3-- N/A

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

   10. lift--.f64 N/A

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

   11. lower-/.f64 91.6

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

   12. lift--.f64 N/A

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

   13. sub-neg N/A

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

   14. lower-+.f64 N/A

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

   15. metadata-eval 91.6

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

4. Applied rewrites 91.6%

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

5. Taylor expanded in z around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. mul-1-neg N/A

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

   5. lower-log1p.f64 N/A

      \[\leadsto \color{blue}{\mathsf{log1p}\left(-1 \cdot y\right)} \cdot z - t \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{neg}\left(y\right)}\right) \cdot z - t \]

   7. lower-neg.f64 45.6

      \[\leadsto \mathsf{log1p}\left(\color{blue}{-y}\right) \cdot z - t \]

7. Applied rewrites 45.6%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(-y\right) \cdot z} - t \]

8. Taylor expanded in y around 0

   \[\leadsto y \cdot \color{blue}{\left(-1 \cdot z + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{3} \cdot \left(y \cdot z\right)\right)\right)} - t \]
9. Step-by-step derivation

10. Applied rewrites 45.6%

    \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(y, z \cdot \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), -z\right)} - t \]
11. Add Preprocessing

Alternative 13: 46.3% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ z \cdot \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), -1\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (* z (* y (fma y (fma y -0.3333333333333333 -0.5) -1.0))) t))

C

double code(double x, double y, double z, double t) {
	return (z * (y * fma(y, fma(y, -0.3333333333333333, -0.5), -1.0))) - t;
}

Julia

function code(x, y, z, t)
	return Float64(Float64(z * Float64(y * fma(y, fma(y, -0.3333333333333333, -0.5), -1.0))) - t)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(z * N[(y * N[(y * N[(y * -0.3333333333333333 + -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 91.7%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. flip3-- N/A

      \[\leadsto \left(\log y \cdot \color{blue}{\frac{{x}^{3} - {1}^{3}}{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   5. clear-num N/A

      \[\leadsto \left(\log y \cdot \color{blue}{\frac{1}{\frac{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}{{x}^{3} - {1}^{3}}}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   6. un-div-inv N/A

      \[\leadsto \left(\color{blue}{\frac{\log y}{\frac{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}{{x}^{3} - {1}^{3}}}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   7. lower-/.f64 N/A

      \[\leadsto \left(\color{blue}{\frac{\log y}{\frac{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}{{x}^{3} - {1}^{3}}}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   8. clear-num N/A

      \[\leadsto \left(\frac{\log y}{\color{blue}{\frac{1}{\frac{{x}^{3} - {1}^{3}}{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}}}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   9. flip3-- N/A

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

   10. lift--.f64 N/A

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

   11. lower-/.f64 91.6

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

   12. lift--.f64 N/A

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

   13. sub-neg N/A

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

   14. lower-+.f64 N/A

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

   15. metadata-eval 91.6

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

4. Applied rewrites 91.6%

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

5. Taylor expanded in z around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. mul-1-neg N/A

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

   5. lower-log1p.f64 N/A

      \[\leadsto \color{blue}{\mathsf{log1p}\left(-1 \cdot y\right)} \cdot z - t \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{neg}\left(y\right)}\right) \cdot z - t \]

   7. lower-neg.f64 45.6

      \[\leadsto \mathsf{log1p}\left(\color{blue}{-y}\right) \cdot z - t \]

7. Applied rewrites 45.6%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(-y\right) \cdot z} - t \]

8. Taylor expanded in y around 0

   \[\leadsto \left(y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right) \cdot z - t \]
9. Step-by-step derivation

10. Applied rewrites 45.6%

    \[\leadsto \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), -1\right)\right) \cdot z - t \]

11. Final simplification 45.6%

    \[\leadsto z \cdot \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), -1\right)\right) - t \]
12. Add Preprocessing

Alternative 14: 43.1% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -125000:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 1200:\\ \;\;\;\;y - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -125000.0) (- t) (if (<= t 1200.0) (- y (* y z)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -125000.0) {
		tmp = -t;
	} else if (t <= 1200.0) {
		tmp = y - (y * z);
	} else {
		tmp = -t;
	}
	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 (t <= (-125000.0d0)) then
        tmp = -t
    else if (t <= 1200.0d0) then
        tmp = y - (y * z)
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -125000.0) {
		tmp = -t;
	} else if (t <= 1200.0) {
		tmp = y - (y * z);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -125000.0:
		tmp = -t
	elif t <= 1200.0:
		tmp = y - (y * z)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -125000.0)
		tmp = Float64(-t);
	elseif (t <= 1200.0)
		tmp = Float64(y - Float64(y * z));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -125000.0)
		tmp = -t;
	elseif (t <= 1200.0)
		tmp = y - (y * z);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -125000.0], (-t), If[LessEqual[t, 1200.0], N[(y - N[(y * z), $MachinePrecision]), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -125000 or 1200 < t

   1. Initial program 96.0%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 73.3

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

   5. Applied rewrites 73.3%

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

   if -125000 < t < 1200

   1. Initial program 87.5%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate--l- N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 14.2%

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

Alternative 15: 42.8% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -125000:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 1200:\\ \;\;\;\;-y \cdot z\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -125000.0) (- t) (if (<= t 1200.0) (- (* y z)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -125000.0) {
		tmp = -t;
	} else if (t <= 1200.0) {
		tmp = -(y * z);
	} else {
		tmp = -t;
	}
	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 (t <= (-125000.0d0)) then
        tmp = -t
    else if (t <= 1200.0d0) then
        tmp = -(y * z)
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -125000.0) {
		tmp = -t;
	} else if (t <= 1200.0) {
		tmp = -(y * z);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -125000.0:
		tmp = -t
	elif t <= 1200.0:
		tmp = -(y * z)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -125000.0)
		tmp = Float64(-t);
	elseif (t <= 1200.0)
		tmp = Float64(-Float64(y * z));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -125000.0)
		tmp = -t;
	elseif (t <= 1200.0)
		tmp = -(y * z);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -125000.0], (-t), If[LessEqual[t, 1200.0], (-N[(y * z), $MachinePrecision]), (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -125000 or 1200 < t

   1. Initial program 96.0%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 73.3

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

   5. Applied rewrites 73.3%

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

   if -125000 < t < 1200

   1. Initial program 87.5%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate--l- N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 13.9%

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

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -125000:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 1200:\\ \;\;\;\;-y \cdot z\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 46.2% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ z \cdot \left(y \cdot \mathsf{fma}\left(y, -0.5, -1\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* z (* y (fma y -0.5 -1.0))) t))

C

double code(double x, double y, double z, double t) {
	return (z * (y * fma(y, -0.5, -1.0))) - t;
}

Julia

function code(x, y, z, t)
	return Float64(Float64(z * Float64(y * fma(y, -0.5, -1.0))) - t)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(z * N[(y * N[(y * -0.5 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 91.7%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. flip3-- N/A

      \[\leadsto \left(\log y \cdot \color{blue}{\frac{{x}^{3} - {1}^{3}}{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   5. clear-num N/A

      \[\leadsto \left(\log y \cdot \color{blue}{\frac{1}{\frac{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}{{x}^{3} - {1}^{3}}}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   6. un-div-inv N/A

      \[\leadsto \left(\color{blue}{\frac{\log y}{\frac{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}{{x}^{3} - {1}^{3}}}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   7. lower-/.f64 N/A

      \[\leadsto \left(\color{blue}{\frac{\log y}{\frac{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}{{x}^{3} - {1}^{3}}}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   8. clear-num N/A

      \[\leadsto \left(\frac{\log y}{\color{blue}{\frac{1}{\frac{{x}^{3} - {1}^{3}}{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}}}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   9. flip3-- N/A

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

   10. lift--.f64 N/A

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

   11. lower-/.f64 91.6

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

   12. lift--.f64 N/A

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

   13. sub-neg N/A

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

   14. lower-+.f64 N/A

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

   15. metadata-eval 91.6

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

4. Applied rewrites 91.6%

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

5. Taylor expanded in z around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. mul-1-neg N/A

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

   5. lower-log1p.f64 N/A

      \[\leadsto \color{blue}{\mathsf{log1p}\left(-1 \cdot y\right)} \cdot z - t \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{neg}\left(y\right)}\right) \cdot z - t \]

   7. lower-neg.f64 45.6

      \[\leadsto \mathsf{log1p}\left(\color{blue}{-y}\right) \cdot z - t \]

7. Applied rewrites 45.6%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(-y\right) \cdot z} - t \]

8. Taylor expanded in y around 0

   \[\leadsto \left(y \cdot \left(\frac{-1}{2} \cdot y - 1\right)\right) \cdot z - t \]
9. Step-by-step derivation

10. Applied rewrites 45.5%

    \[\leadsto \left(y \cdot \mathsf{fma}\left(y, -0.5, -1\right)\right) \cdot z - t \]

11. Final simplification 45.5%

    \[\leadsto z \cdot \left(y \cdot \mathsf{fma}\left(y, -0.5, -1\right)\right) - t \]
12. Add Preprocessing

Alternative 17: 45.9% accurate, 20.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.7%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. flip3-- N/A

      \[\leadsto \left(\log y \cdot \color{blue}{\frac{{x}^{3} - {1}^{3}}{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   5. clear-num N/A

      \[\leadsto \left(\log y \cdot \color{blue}{\frac{1}{\frac{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}{{x}^{3} - {1}^{3}}}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   6. un-div-inv N/A

      \[\leadsto \left(\color{blue}{\frac{\log y}{\frac{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}{{x}^{3} - {1}^{3}}}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   7. lower-/.f64 N/A

      \[\leadsto \left(\color{blue}{\frac{\log y}{\frac{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}{{x}^{3} - {1}^{3}}}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   8. clear-num N/A

      \[\leadsto \left(\frac{\log y}{\color{blue}{\frac{1}{\frac{{x}^{3} - {1}^{3}}{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}}}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   9. flip3-- N/A

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

   10. lift--.f64 N/A

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

   11. lower-/.f64 91.6

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

   12. lift--.f64 N/A

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

   13. sub-neg N/A

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

   14. lower-+.f64 N/A

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

   15. metadata-eval 91.6

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

4. Applied rewrites 91.6%

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

5. Taylor expanded in z around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. mul-1-neg N/A

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

   5. lower-log1p.f64 N/A

      \[\leadsto \color{blue}{\mathsf{log1p}\left(-1 \cdot y\right)} \cdot z - t \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{neg}\left(y\right)}\right) \cdot z - t \]

   7. lower-neg.f64 45.6

      \[\leadsto \mathsf{log1p}\left(\color{blue}{-y}\right) \cdot z - t \]

7. Applied rewrites 45.6%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(-y\right) \cdot z} - t \]

8. Taylor expanded in y around 0

   \[\leadsto \left(-1 \cdot y\right) \cdot z - t \]
9. Step-by-step derivation

10. Applied rewrites 45.4%

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

11. Final simplification 45.4%

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

Alternative 18: 35.6% accurate, 75.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.7%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 37.8

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

5. Applied rewrites 37.8%

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

Reproduce

herbie shell --seed 2024220 
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  :precision binary64
  (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))