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

Percentage Accurate: 89.1% → 99.6%

Time: 18.6s

Alternatives: 15

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: 89.1% 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

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (+
   (* (+ x -1.0) (log y))
   (*
    (+ z -1.0)
    (* 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 (((x + -1.0) * log(y)) + ((z + -1.0) * (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(Float64(Float64(x + -1.0) * log(y)) + Float64(Float64(z + -1.0) * 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[(N[(N[(x + -1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] + 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]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 91.4%

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

3. Taylor expanded in y around 0

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}\right) - t \]
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 99.8

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

5. Applied rewrites 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x + -1.0) * log(y)) + Float64(Float64(z + -1.0) * Float64(y * fma(y, fma(y, -0.3333333333333333, -0.5), -1.0)))) - 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[(y * N[(y * N[(y * -0.3333333333333333 + -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 91.4%

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

3. Taylor expanded in y around 0

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 99.7

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

5. Applied rewrites 99.7%

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

6. Final simplification 99.7%

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

Alternative 3: 95.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 1 - z, x \cdot \log y\right) - t\\ \mathbf{if}\;z + -1 \leq -2 \cdot 10^{+138}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z + -1 \leq 5 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x + -1, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (fma y (- 1.0 z) (* x (log y))) t)))
   (if (<= (+ z -1.0) -2e+138)
     t_1
     (if (<= (+ z -1.0) 5e+147) (- (fma (log y) (+ x -1.0) y) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(y, (1.0 - z), (x * log(y))) - t;
	double tmp;
	if ((z + -1.0) <= -2e+138) {
		tmp = t_1;
	} else if ((z + -1.0) <= 5e+147) {
		tmp = fma(log(y), (x + -1.0), y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(y, Float64(1.0 - z), Float64(x * log(y))) - t)
	tmp = 0.0
	if (Float64(z + -1.0) <= -2e+138)
		tmp = t_1;
	elseif (Float64(z + -1.0) <= 5e+147)
		tmp = Float64(fma(log(y), Float64(x + -1.0), y) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * N[(1.0 - z), $MachinePrecision] + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[N[(z + -1.0), $MachinePrecision], -2e+138], t$95$1, If[LessEqual[N[(z + -1.0), $MachinePrecision], 5e+147], N[(N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision] + y), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z #s(literal 1 binary64)) < -2.0000000000000001e138 or 5.0000000000000002e147 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 65.6%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{0 - \left(z - 1\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

      7. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(y, 0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(1\right)\right)\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(y, 0 - \left(z + \color{blue}{-1}\right), \log y \cdot \left(x - 1\right)\right) - t \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, 0 - \color{blue}{\left(-1 + z\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(0 - -1\right) - z}, \log y \cdot \left(x - 1\right)\right) - t \]

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 93.6%

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

   if -2.0000000000000001e138 < (-.f64 z #s(literal 1 binary64)) < 5.0000000000000002e147

   1. Initial program 99.2%

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

   3. Taylor expanded in 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. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{0 - \left(z - 1\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

      7. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(y, 0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(1\right)\right)\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(y, 0 - \left(z + \color{blue}{-1}\right), \log y \cdot \left(x - 1\right)\right) - t \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, 0 - \color{blue}{\left(-1 + z\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(0 - -1\right) - z}, \log y \cdot \left(x - 1\right)\right) - t \]

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 99.0%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + -1 \leq -2 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - z, x \cdot \log y\right) - t\\ \mathbf{elif}\;z + -1 \leq 5 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x + -1, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - z, x \cdot \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

5. Applied rewrites 99.7%

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

6. Final simplification 99.7%

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

Alternative 5: 77.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;x + -1 \leq -2 \cdot 10^{+24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x + -1 \leq -0.9998:\\ \;\;\;\;\left(z + -1\right) \cdot \left(y \cdot \mathsf{fma}\left(y, -0.5, -1\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t)))
   (if (<= (+ x -1.0) -2e+24)
     t_1
     (if (<= (+ x -1.0) -0.9998)
       (- (* (+ z -1.0) (* y (fma y -0.5 -1.0))) t)
       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) <= -2e+24) {
		tmp = t_1;
	} else if ((x + -1.0) <= -0.9998) {
		tmp = ((z + -1.0) * (y * fma(y, -0.5, -1.0))) - t;
	} 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) <= -2e+24)
		tmp = t_1;
	elseif (Float64(x + -1.0) <= -0.9998)
		tmp = Float64(Float64(Float64(z + -1.0) * Float64(y * fma(y, -0.5, -1.0))) - t);
	else
		tmp = t_1;
	end
	return 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], -2e+24], t$95$1, If[LessEqual[N[(x + -1.0), $MachinePrecision], -0.9998], N[(N[(N[(z + -1.0), $MachinePrecision] * N[(y * N[(y * -0.5 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -2e24 or -0.99980000000000002 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 98.0%

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

   3. Taylor expanded in 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 98.0

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

   5. Applied rewrites 98.0%

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

   if -2e24 < (-.f64 x #s(literal 1 binary64)) < -0.99980000000000002

   1. Initial program 84.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 41.4%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 62.4%

       \[\leadsto \left(-1 + z\right) \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(y, -0.5, -1\right)}\right) - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + -1 \leq -2 \cdot 10^{+24}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x + -1 \leq -0.9998:\\ \;\;\;\;\left(z + -1\right) \cdot \left(y \cdot \mathsf{fma}\left(y, -0.5, -1\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x + -1 \leq -2 \cdot 10^{+24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x + -1 \leq 10^{+19}:\\ \;\;\;\;\left(z + -1\right) \cdot \left(y \cdot \mathsf{fma}\left(y, -0.5, -1\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= (+ x -1.0) -2e+24)
     t_1
     (if (<= (+ x -1.0) 1e+19)
       (- (* (+ z -1.0) (* y (fma y -0.5 -1.0))) 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) <= -2e+24) {
		tmp = t_1;
	} else if ((x + -1.0) <= 1e+19) {
		tmp = ((z + -1.0) * (y * fma(y, -0.5, -1.0))) - 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) <= -2e+24)
		tmp = t_1;
	elseif (Float64(x + -1.0) <= 1e+19)
		tmp = Float64(Float64(Float64(z + -1.0) * Float64(y * fma(y, -0.5, -1.0))) - t);
	else
		tmp = t_1;
	end
	return 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], -2e+24], t$95$1, If[LessEqual[N[(x + -1.0), $MachinePrecision], 1e+19], N[(N[(N[(z + -1.0), $MachinePrecision] * N[(y * N[(y * -0.5 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -2e24 or 1e19 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 98.0%

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

   3. Taylor expanded in 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 75.8

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

   5. Applied rewrites 75.8%

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

   if -2e24 < (-.f64 x #s(literal 1 binary64)) < 1e19

   1. Initial program 84.6%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 42.0%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 63.3%

       \[\leadsto \left(-1 + z\right) \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(y, -0.5, -1\right)}\right) - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + -1 \leq -2 \cdot 10^{+24}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x + -1 \leq 10^{+19}:\\ \;\;\;\;\left(z + -1\right) \cdot \left(y \cdot \mathsf{fma}\left(y, -0.5, -1\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z + -1 \leq 5 \cdot 10^{+244}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x + -1, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(y - y \cdot z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ z -1.0) 5e+244)
   (- (fma (log y) (+ x -1.0) y) t)
   (- (- y (* y z)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z + -1.0) <= 5e+244) {
		tmp = fma(log(y), (x + -1.0), y) - t;
	} else {
		tmp = (y - (y * z)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z + -1.0) <= 5e+244)
		tmp = Float64(fma(log(y), Float64(x + -1.0), y) - t);
	else
		tmp = Float64(Float64(y - Float64(y * z)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z + -1.0), $MachinePrecision], 5e+244], N[(N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision] + y), $MachinePrecision] - t), $MachinePrecision], N[(N[(y - N[(y * z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z #s(literal 1 binary64)) < 5.00000000000000022e244

   1. Initial program 94.2%

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

   3. Taylor expanded in 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. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{0 - \left(z - 1\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

      7. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(y, 0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(1\right)\right)\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(y, 0 - \left(z + \color{blue}{-1}\right), \log y \cdot \left(x - 1\right)\right) - t \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, 0 - \color{blue}{\left(-1 + z\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(0 - -1\right) - z}, \log y \cdot \left(x - 1\right)\right) - t \]

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 94.0%

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

   if 5.00000000000000022e244 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 28.1%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{0 - \left(z - 1\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

      7. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(y, 0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(1\right)\right)\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(y, 0 - \left(z + \color{blue}{-1}\right), \log y \cdot \left(x - 1\right)\right) - t \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, 0 - \color{blue}{\left(-1 + z\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(0 - -1\right) - z}, \log y \cdot \left(x - 1\right)\right) - t \]

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 88.3%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + -1 \leq 5 \cdot 10^{+244}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x + -1, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(y - y \cdot z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z + -1 \leq 5 \cdot 10^{+244}:\\ \;\;\;\;\left(x + -1\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(y - y \cdot z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ z -1.0) 5e+244) (- (* (+ x -1.0) (log y)) t) (- (- y (* y z)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z + -1.0) <= 5e+244) {
		tmp = ((x + -1.0) * log(y)) - t;
	} else {
		tmp = (y - (y * z)) - 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+244) then
        tmp = ((x + (-1.0d0)) * log(y)) - t
    else
        tmp = (y - (y * z)) - 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+244) {
		tmp = ((x + -1.0) * Math.log(y)) - t;
	} else {
		tmp = (y - (y * z)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z + -1.0) <= 5e+244:
		tmp = ((x + -1.0) * math.log(y)) - t
	else:
		tmp = (y - (y * z)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z + -1.0) <= 5e+244)
		tmp = Float64(Float64(Float64(x + -1.0) * log(y)) - t);
	else
		tmp = Float64(Float64(y - Float64(y * z)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z + -1.0) <= 5e+244)
		tmp = ((x + -1.0) * log(y)) - t;
	else
		tmp = (y - (y * z)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z + -1.0), $MachinePrecision], 5e+244], N[(N[(N[(x + -1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(y - N[(y * z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z #s(literal 1 binary64)) < 5.00000000000000022e244

   1. Initial program 94.2%

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

   3. Taylor expanded in 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 5.00000000000000022e244 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 28.1%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{0 - \left(z - 1\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

      7. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(y, 0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(1\right)\right)\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(y, 0 - \left(z + \color{blue}{-1}\right), \log y \cdot \left(x - 1\right)\right) - t \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, 0 - \color{blue}{\left(-1 + z\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(0 - -1\right) - z}, \log y \cdot \left(x - 1\right)\right) - t \]

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 88.3%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + -1 \leq 5 \cdot 10^{+244}:\\ \;\;\;\;\left(x + -1\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(y - y \cdot z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.4%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
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. mul-1-neg N/A

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

   2. distribute-rgt-neg-in N/A

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

   3. mul-1-neg N/A

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

   4. lower-fma.f64 N/A

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

   5. mul-1-neg N/A

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

   6. neg-sub0 N/A

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{0 - \left(z - 1\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

   7. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(y, 0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(1\right)\right)\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(y, 0 - \left(z + \color{blue}{-1}\right), \log y \cdot \left(x - 1\right)\right) - t \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(y, 0 - \color{blue}{\left(-1 + z\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

   10. associate--r+ N/A

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(0 - -1\right) - z}, \log y \cdot \left(x - 1\right)\right) - t \]

   11. metadata-eval N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-log.f64 N/A

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

   15. sub-neg N/A

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

   16. metadata-eval N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 99.6

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

5. Applied rewrites 99.6%

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

6. Final simplification 99.6%

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

Alternative 10: 42.2% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{+33}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{+60}:\\ \;\;\;\;t + z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2e+33) (- t) (if (<= t 2.65e+60) (+ t (* z (- y))) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2e+33) {
		tmp = -t;
	} else if (t <= 2.65e+60) {
		tmp = t + (z * -y);
	} 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 <= (-2d+33)) then
        tmp = -t
    else if (t <= 2.65d+60) then
        tmp = t + (z * -y)
    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 <= -2e+33) {
		tmp = -t;
	} else if (t <= 2.65e+60) {
		tmp = t + (z * -y);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -2e+33:
		tmp = -t
	elif t <= 2.65e+60:
		tmp = t + (z * -y)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2e+33)
		tmp = Float64(-t);
	elseif (t <= 2.65e+60)
		tmp = Float64(t + Float64(z * Float64(-y)));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -2e+33)
		tmp = -t;
	elseif (t <= 2.65e+60)
		tmp = t + (z * -y);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2e+33], (-t), If[LessEqual[t, 2.65e+60], N[(t + N[(z * (-y)), $MachinePrecision]), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -1.9999999999999999e33 or 2.6499999999999998e60 < t

   1. Initial program 99.0%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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 76.9

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

   5. Applied rewrites 76.9%

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

   if -1.9999999999999999e33 < t < 2.6499999999999998e60

   1. Initial program 85.6%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \log \left(1 - y\right)} - t \]
   4. 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. lower-log1p.f64 N/A

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

      5. lower-neg.f64 18.2

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

   5. Applied rewrites 18.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 18.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. unpow1 N/A

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

      4. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(y\right)\right) \cdot z + \left(\mathsf{neg}\left({t}^{\color{blue}{\left(3 - 2\right)}}\right)\right) \]

      5. pow-div N/A

         \[\leadsto \left(\mathsf{neg}\left(y\right)\right) \cdot z + \left(\mathsf{neg}\left(\color{blue}{\frac{{t}^{3}}{{t}^{2}}}\right)\right) \]

      6. cube-unmult N/A

         \[\leadsto \left(\mathsf{neg}\left(y\right)\right) \cdot z + \left(\mathsf{neg}\left(\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{{t}^{2}}\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(y\right)\right) \cdot z + \left(\mathsf{neg}\left(\frac{t \cdot \color{blue}{\left(t \cdot t\right)}}{{t}^{2}}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(y\right)\right) \cdot z + \left(\mathsf{neg}\left(\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{{t}^{2}}\right)\right) \]

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-frac-neg N/A

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

   10. Applied rewrites 16.9%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{+33}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{+60}:\\ \;\;\;\;t + z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.0% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 31.1%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 44.0%

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

12. Final simplification 44.0%

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

Alternative 12: 46.7% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.4%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
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. mul-1-neg N/A

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

   2. distribute-rgt-neg-in N/A

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

   3. mul-1-neg N/A

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

   4. lower-fma.f64 N/A

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

   5. mul-1-neg N/A

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

   6. neg-sub0 N/A

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{0 - \left(z - 1\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

   7. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(y, 0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(1\right)\right)\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(y, 0 - \left(z + \color{blue}{-1}\right), \log y \cdot \left(x - 1\right)\right) - t \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(y, 0 - \color{blue}{\left(-1 + z\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

   10. associate--r+ N/A

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(0 - -1\right) - z}, \log y \cdot \left(x - 1\right)\right) - t \]

   11. metadata-eval N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-log.f64 N/A

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

   15. sub-neg N/A

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

   16. metadata-eval N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 99.6

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 44.0%

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

Alternative 13: 46.5% 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 91.4%

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

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{z \cdot \log \left(1 - y\right)} - t \]
4. 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. lower-log1p.f64 N/A

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

   5. lower-neg.f64 43.8

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

5. Applied rewrites 43.8%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 43.8%

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

9. Final simplification 43.8%

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

Alternative 14: 36.0% 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 91.4%

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

3. Taylor expanded in 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 35.5

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

5. Applied rewrites 35.5%

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

Alternative 15: 2.3% accurate, 226.0× 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 t
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := t

Derivation

1. Initial program 91.4%

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

3. Taylor expanded in 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 35.5

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

5. Applied rewrites 35.5%

   \[\leadsto \color{blue}{-t} \]
6. Step-by-step derivation

7. Applied rewrites 6.6%

   \[\leadsto \frac{0 - t \cdot \left(t \cdot t\right)}{\color{blue}{0 + \mathsf{fma}\left(t, t, 0 \cdot t\right)}} \]
8. Step-by-step derivation

9. Applied rewrites 2.3%

   \[\leadsto t \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(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))