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

Percentage Accurate: 88.6% → 99.5%

Time: 12.6s

Alternatives: 16

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: 88.6% 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: 99.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

3. Taylor expanded in y around 0

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

   1. associate--l+ N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 73.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ t_2 := \log \left(1 - y\right) \cdot \left(z - 1\right) - \left(1 - x\right) \cdot \log y\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+124}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 350:\\ \;\;\;\;\mathsf{fma}\left(-z, y, y\right) - t\\ \mathbf{elif}\;t\_2 \leq 4000000000000:\\ \;\;\;\;\left(y - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x))
        (t_2 (- (* (log (- 1.0 y)) (- z 1.0)) (* (- 1.0 x) (log y)))))
   (if (<= t_2 -2e+124)
     t_1
     (if (<= t_2 350.0)
       (- (fma (- z) y y) t)
       (if (<= t_2 4000000000000.0) (- (- y (log y)) t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double t_2 = (log((1.0 - y)) * (z - 1.0)) - ((1.0 - x) * log(y));
	double tmp;
	if (t_2 <= -2e+124) {
		tmp = t_1;
	} else if (t_2 <= 350.0) {
		tmp = fma(-z, y, y) - t;
	} else if (t_2 <= 4000000000000.0) {
		tmp = (y - log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	t_2 = Float64(Float64(log(Float64(1.0 - y)) * Float64(z - 1.0)) - Float64(Float64(1.0 - x) * log(y)))
	tmp = 0.0
	if (t_2 <= -2e+124)
		tmp = t_1;
	elseif (t_2 <= 350.0)
		tmp = Float64(fma(Float64(-z), y, y) - t);
	elseif (t_2 <= 4000000000000.0)
		tmp = Float64(Float64(y - log(y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision] * N[(z - 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 - x), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+124], t$95$1, If[LessEqual[t$95$2, 350.0], N[(N[((-z) * y + y), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$2, 4000000000000.0], N[(N[(y - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.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)))) < -1.9999999999999999e124 or 4e12 < (+.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))))

   1. Initial program 95.7%

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

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 79.9

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

   8. Applied rewrites 79.9%

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

   if -1.9999999999999999e124 < (+.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)))) < 350

   1. Initial program 71.4%

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

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. mul-1-neg N/A

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-in N/A

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

      11. neg-mul-1 N/A

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

      12. metadata-eval N/A

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. lower-fma.f64 N/A

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

      16. neg-mul-1 N/A

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

      17. lower-neg.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 64.4%

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

   if 350 < (+.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)))) < 4e12

   1. Initial program 89.1%

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

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. mul-1-neg N/A

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-in N/A

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

      11. neg-mul-1 N/A

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

      12. metadata-eval N/A

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. lower-fma.f64 N/A

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

      16. neg-mul-1 N/A

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

      17. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.0%

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

   9. Taylor expanded in z around 0

      \[\leadsto \left(y - \log y\right) - t \]
   10. Step-by-step derivation

   11. Applied rewrites 87.1%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 - y\right) \cdot \left(z - 1\right) - \left(1 - x\right) \cdot \log y \leq -2 \cdot 10^{+124}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;\log \left(1 - y\right) \cdot \left(z - 1\right) - \left(1 - x\right) \cdot \log y \leq 350:\\ \;\;\;\;\mathsf{fma}\left(-z, y, y\right) - t\\ \mathbf{elif}\;\log \left(1 - y\right) \cdot \left(z - 1\right) - \left(1 - x\right) \cdot \log y \leq 4000000000000:\\ \;\;\;\;\left(y - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ t_2 := \log \left(1 - y\right) \cdot \left(z - 1\right) - \left(1 - x\right) \cdot \log y\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+124}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 350:\\ \;\;\;\;\mathsf{fma}\left(-z, y, y\right) - t\\ \mathbf{elif}\;t\_2 \leq 4000000000000:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x))
        (t_2 (- (* (log (- 1.0 y)) (- z 1.0)) (* (- 1.0 x) (log y)))))
   (if (<= t_2 -2e+124)
     t_1
     (if (<= t_2 350.0)
       (- (fma (- z) y y) t)
       (if (<= t_2 4000000000000.0) (- (- (log y)) t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double t_2 = (log((1.0 - y)) * (z - 1.0)) - ((1.0 - x) * log(y));
	double tmp;
	if (t_2 <= -2e+124) {
		tmp = t_1;
	} else if (t_2 <= 350.0) {
		tmp = fma(-z, y, y) - t;
	} else if (t_2 <= 4000000000000.0) {
		tmp = -log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	t_2 = Float64(Float64(log(Float64(1.0 - y)) * Float64(z - 1.0)) - Float64(Float64(1.0 - x) * log(y)))
	tmp = 0.0
	if (t_2 <= -2e+124)
		tmp = t_1;
	elseif (t_2 <= 350.0)
		tmp = Float64(fma(Float64(-z), y, y) - t);
	elseif (t_2 <= 4000000000000.0)
		tmp = Float64(Float64(-log(y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision] * N[(z - 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 - x), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+124], t$95$1, If[LessEqual[t$95$2, 350.0], N[(N[((-z) * y + y), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$2, 4000000000000.0], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.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)))) < -1.9999999999999999e124 or 4e12 < (+.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))))

   1. Initial program 95.7%

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

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 79.9

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

   8. Applied rewrites 79.9%

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

   if -1.9999999999999999e124 < (+.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)))) < 350

   1. Initial program 71.4%

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

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. mul-1-neg N/A

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-in N/A

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

      11. neg-mul-1 N/A

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

      12. metadata-eval N/A

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. lower-fma.f64 N/A

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

      16. neg-mul-1 N/A

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

      17. lower-neg.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 64.4%

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

   if 350 < (+.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)))) < 4e12

   1. Initial program 89.1%

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

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. mul-1-neg N/A

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-in N/A

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

      11. neg-mul-1 N/A

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

      12. metadata-eval N/A

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. lower-fma.f64 N/A

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

      16. neg-mul-1 N/A

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

      17. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.0%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 87.1%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 - y\right) \cdot \left(z - 1\right) - \left(1 - x\right) \cdot \log y \leq -2 \cdot 10^{+124}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;\log \left(1 - y\right) \cdot \left(z - 1\right) - \left(1 - x\right) \cdot \log y \leq 350:\\ \;\;\;\;\mathsf{fma}\left(-z, y, y\right) - t\\ \mathbf{elif}\;\log \left(1 - y\right) \cdot \left(z - 1\right) - \left(1 - x\right) \cdot \log y \leq 4000000000000:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x - t\\ t_2 := \left(\log \left(1 - y\right) \cdot \left(z - 1\right) - \left(1 - x\right) \cdot \log y\right) - t\\ \mathbf{if}\;t\_2 \leq -10000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2000:\\ \;\;\;\;\left(y - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (log y) x) t))
        (t_2 (- (- (* (log (- 1.0 y)) (- z 1.0)) (* (- 1.0 x) (log y))) t)))
   (if (<= t_2 -10000000000000.0)
     t_1
     (if (<= t_2 2000.0) (- (- y (log y)) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (log(y) * x) - t;
	double t_2 = ((log((1.0 - y)) * (z - 1.0)) - ((1.0 - x) * log(y))) - t;
	double tmp;
	if (t_2 <= -10000000000000.0) {
		tmp = t_1;
	} else if (t_2 <= 2000.0) {
		tmp = (y - log(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) :: t_2
    real(8) :: tmp
    t_1 = (log(y) * x) - t
    t_2 = ((log((1.0d0 - y)) * (z - 1.0d0)) - ((1.0d0 - x) * log(y))) - t
    if (t_2 <= (-10000000000000.0d0)) then
        tmp = t_1
    else if (t_2 <= 2000.0d0) then
        tmp = (y - log(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 = (Math.log(y) * x) - t;
	double t_2 = ((Math.log((1.0 - y)) * (z - 1.0)) - ((1.0 - x) * Math.log(y))) - t;
	double tmp;
	if (t_2 <= -10000000000000.0) {
		tmp = t_1;
	} else if (t_2 <= 2000.0) {
		tmp = (y - Math.log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (math.log(y) * x) - t
	t_2 = ((math.log((1.0 - y)) * (z - 1.0)) - ((1.0 - x) * math.log(y))) - t
	tmp = 0
	if t_2 <= -10000000000000.0:
		tmp = t_1
	elif t_2 <= 2000.0:
		tmp = (y - math.log(y)) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(log(y) * x) - t)
	t_2 = Float64(Float64(Float64(log(Float64(1.0 - y)) * Float64(z - 1.0)) - Float64(Float64(1.0 - x) * log(y))) - t)
	tmp = 0.0
	if (t_2 <= -10000000000000.0)
		tmp = t_1;
	elseif (t_2 <= 2000.0)
		tmp = Float64(Float64(y - log(y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (log(y) * x) - t;
	t_2 = ((log((1.0 - y)) * (z - 1.0)) - ((1.0 - x) * log(y))) - t;
	tmp = 0.0;
	if (t_2 <= -10000000000000.0)
		tmp = t_1;
	elseif (t_2 <= 2000.0)
		tmp = (y - log(y)) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision] * N[(z - 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 - x), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$2, -10000000000000.0], t$95$1, If[LessEqual[t$95$2, 2000.0], N[(N[(y - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]]

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) < -1e13 or 2e3 < (-.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 89.7%

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

   3. Taylor expanded in 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 88.5

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

   5. Applied rewrites 88.5%

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

   if -1e13 < (-.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) < 2e3

   1. Initial program 72.2%

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

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. mul-1-neg N/A

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-in N/A

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

      11. neg-mul-1 N/A

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

      12. metadata-eval N/A

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. lower-fma.f64 N/A

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

      16. neg-mul-1 N/A

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

      17. lower-neg.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.5%

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

   9. Taylor expanded in z around 0

      \[\leadsto \left(y - \log y\right) - t \]
   10. Step-by-step derivation

   11. Applied rewrites 69.3%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\log \left(1 - y\right) \cdot \left(z - 1\right) - \left(1 - x\right) \cdot \log y\right) - t \leq -10000000000000:\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{elif}\;\left(\log \left(1 - y\right) \cdot \left(z - 1\right) - \left(1 - x\right) \cdot \log y\right) - t \leq 2000:\\ \;\;\;\;\left(y - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 99.7

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

5. Applied rewrites 99.7%

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

6. Final simplification 99.7%

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

Alternative 6: 95.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - 1, \log y, -t\right)\\ \mathbf{if}\;x - 1 \leq -300000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x - 1 \leq -0.99999999999999:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, -\left(\log y + t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (- x 1.0) (log y) (- t))))
   (if (<= (- x 1.0) -300000000.0)
     t_1
     (if (<= (- x 1.0) -0.99999999999999)
       (fma (- 1.0 z) y (- (+ (log y) t)))
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((x - 1.0), log(y), -t);
	double tmp;
	if ((x - 1.0) <= -300000000.0) {
		tmp = t_1;
	} else if ((x - 1.0) <= -0.99999999999999) {
		tmp = fma((1.0 - z), y, -(log(y) + t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(x - 1.0), log(y), Float64(-t))
	tmp = 0.0
	if (Float64(x - 1.0) <= -300000000.0)
		tmp = t_1;
	elseif (Float64(x - 1.0) <= -0.99999999999999)
		tmp = fma(Float64(1.0 - z), y, Float64(-Float64(log(y) + t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision]}, If[LessEqual[N[(x - 1.0), $MachinePrecision], -300000000.0], t$95$1, If[LessEqual[N[(x - 1.0), $MachinePrecision], -0.99999999999999], N[(N[(1.0 - z), $MachinePrecision] * y + (-N[(N[Log[y], $MachinePrecision] + t), $MachinePrecision])), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -3e8 or -0.99999999999999001 < (-.f64 x #s(literal 1 binary64))

   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

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-neg.f64 91.2

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

   5. Applied rewrites 91.2%

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

   if -3e8 < (-.f64 x #s(literal 1 binary64)) < -0.99999999999999001

   1. Initial program 78.8%

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

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. distribute-lft-in N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 N/A

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

      15. sub-neg N/A

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. distribute-rgt-in N/A

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

      19. +-commutative N/A

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

      20. distribute-rgt-out N/A

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

      21. lower-fma.f64 N/A

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

   8. Applied rewrites 99.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 98.6%

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

Alternative 7: 99.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

3. Taylor expanded in y around 0

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

   1. distribute-lft-in N/A

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

   2. mul-1-neg N/A

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

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

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

   4. mul-1-neg N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. distribute-rgt-out N/A

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

   8. +-commutative N/A

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

   9. metadata-eval N/A

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

   10. sub-neg N/A

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

   11. lower-fma.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower--.f64 N/A

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

   15. sub-neg N/A

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

   16. metadata-eval N/A

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

   17. lower-fma.f64 N/A

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

   18. *-commutative N/A

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

5. Applied rewrites 99.6%

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

6. Final simplification 99.6%

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

Alternative 8: 95.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - 1, \log y, -t\right)\\ \mathbf{if}\;x - 1 \leq -300000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x - 1 \leq -0.99999999999999:\\ \;\;\;\;\left(y - \mathsf{fma}\left(y, z, \log y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (- x 1.0) (log y) (- t))))
   (if (<= (- x 1.0) -300000000.0)
     t_1
     (if (<= (- x 1.0) -0.99999999999999) (- (- y (fma y z (log y))) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((x - 1.0), log(y), -t);
	double tmp;
	if ((x - 1.0) <= -300000000.0) {
		tmp = t_1;
	} else if ((x - 1.0) <= -0.99999999999999) {
		tmp = (y - fma(y, z, log(y))) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(x - 1.0), log(y), Float64(-t))
	tmp = 0.0
	if (Float64(x - 1.0) <= -300000000.0)
		tmp = t_1;
	elseif (Float64(x - 1.0) <= -0.99999999999999)
		tmp = Float64(Float64(y - fma(y, z, log(y))) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision]}, If[LessEqual[N[(x - 1.0), $MachinePrecision], -300000000.0], t$95$1, If[LessEqual[N[(x - 1.0), $MachinePrecision], -0.99999999999999], N[(N[(y - N[(y * z + N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -3e8 or -0.99999999999999001 < (-.f64 x #s(literal 1 binary64))

   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

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-neg.f64 91.2

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

   5. Applied rewrites 91.2%

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

   if -3e8 < (-.f64 x #s(literal 1 binary64)) < -0.99999999999999001

   1. Initial program 78.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. mul-1-neg N/A

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-in N/A

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

      11. neg-mul-1 N/A

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

      12. metadata-eval N/A

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. lower-fma.f64 N/A

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

      16. neg-mul-1 N/A

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

      17. lower-neg.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 64.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 98.6%

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

Alternative 9: 88.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{log1p}\left(-y\right) \cdot z - t\\ \mathbf{if}\;z - 1 \leq -1 \cdot 10^{+247}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z - 1 \leq 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (log1p (- y)) z) t)))
   (if (<= (- z 1.0) -1e+247)
     t_1
     (if (<= (- z 1.0) 1e+154) (fma (- x 1.0) (log y) (- t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (log1p(-y) * z) - t;
	double tmp;
	if ((z - 1.0) <= -1e+247) {
		tmp = t_1;
	} else if ((z - 1.0) <= 1e+154) {
		tmp = fma((x - 1.0), log(y), -t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(log1p(Float64(-y)) * z) - t)
	tmp = 0.0
	if (Float64(z - 1.0) <= -1e+247)
		tmp = t_1;
	elseif (Float64(z - 1.0) <= 1e+154)
		tmp = fma(Float64(x - 1.0), log(y), Float64(-t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Log[1 + (-y)], $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[N[(z - 1.0), $MachinePrecision], -1e+247], t$95$1, If[LessEqual[N[(z - 1.0), $MachinePrecision], 1e+154], N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z #s(literal 1 binary64)) < -9.99999999999999952e246 or 1.00000000000000004e154 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 42.6%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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 78.8

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

   5. Applied rewrites 78.8%

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

   if -9.99999999999999952e246 < (-.f64 z #s(literal 1 binary64)) < 1.00000000000000004e154

   1. Initial program 95.6%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-neg.f64 95.2

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

   5. Applied rewrites 95.2%

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

Alternative 10: 66.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x - 1 \leq -5 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x - 1 \leq 2 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= (- x 1.0) -5e+20)
     t_1
     (if (<= (- x 1.0) 2e+120) (- (fma (- z) y y) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if ((x - 1.0) <= -5e+20) {
		tmp = t_1;
	} else if ((x - 1.0) <= 2e+120) {
		tmp = fma(-z, y, y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (Float64(x - 1.0) <= -5e+20)
		tmp = t_1;
	elseif (Float64(x - 1.0) <= 2e+120)
		tmp = Float64(fma(Float64(-z), y, y) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(x - 1.0), $MachinePrecision], -5e+20], t$95$1, If[LessEqual[N[(x - 1.0), $MachinePrecision], 2e+120], N[(N[((-z) * y + y), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -5e20 or 2e120 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 96.5%

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

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 81.8

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

   8. Applied rewrites 81.8%

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

   if -5e20 < (-.f64 x #s(literal 1 binary64)) < 2e120

   1. Initial program 79.2%

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

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. mul-1-neg N/A

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-in N/A

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

      11. neg-mul-1 N/A

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

      12. metadata-eval N/A

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. lower-fma.f64 N/A

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

      16. neg-mul-1 N/A

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

      17. lower-neg.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 62.7%

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

Alternative 11: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

3. Taylor expanded in y around 0

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

   1. associate--l+ N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in y around 0

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

   1. associate--l+ N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

   4. distribute-lft-neg-in N/A

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

   5. mul-1-neg N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. distribute-lft-in N/A

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

   10. metadata-eval N/A

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

   11. +-commutative N/A

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

   12. mul-1-neg N/A

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

   13. unsub-neg N/A

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

   14. lower--.f64 N/A

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

   15. sub-neg N/A

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

   16. sub-neg N/A

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

   17. metadata-eval N/A

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

   18. distribute-rgt-in N/A

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

   19. +-commutative N/A

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

   20. distribute-rgt-out N/A

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

   21. lower-fma.f64 N/A

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

8. Applied rewrites 99.4%

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

9. Final simplification 99.4%

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

Alternative 12: 43.4% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{-11}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 62000000:\\ \;\;\;\;y - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.05e-11) (- t) (if (<= t 62000000.0) (- y (* z y)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.05e-11) {
		tmp = -t;
	} else if (t <= 62000000.0) {
		tmp = y - (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 <= (-1.05d-11)) then
        tmp = -t
    else if (t <= 62000000.0d0) then
        tmp = y - (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 <= -1.05e-11) {
		tmp = -t;
	} else if (t <= 62000000.0) {
		tmp = y - (z * y);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.05e-11:
		tmp = -t
	elif t <= 62000000.0:
		tmp = y - (z * y)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.05e-11)
		tmp = Float64(-t);
	elseif (t <= 62000000.0)
		tmp = Float64(y - Float64(z * y));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.05e-11)
		tmp = -t;
	elseif (t <= 62000000.0)
		tmp = y - (z * y);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.05e-11], (-t), If[LessEqual[t, 62000000.0], N[(y - N[(z * y), $MachinePrecision]), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -1.0499999999999999e-11 or 6.2e7 < t

   1. Initial program 91.8%

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

   3. Taylor expanded in 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 68.0

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

   5. Applied rewrites 68.0%

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

   if -1.0499999999999999e-11 < t < 6.2e7

   1. Initial program 78.9%

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

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. distribute-lft-in N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 N/A

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

      15. sub-neg N/A

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. distribute-rgt-in N/A

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

      19. +-commutative N/A

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

      20. distribute-rgt-out N/A

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

      21. lower-fma.f64 N/A

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

   8. Applied rewrites 99.4%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 24.9%

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

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{-11}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 62000000:\\ \;\;\;\;y - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 43.1% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{-11}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 62000000:\\ \;\;\;\;\left(-y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.05e-11) (- t) (if (<= t 62000000.0) (* (- y) z) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.05e-11) {
		tmp = -t;
	} else if (t <= 62000000.0) {
		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 <= (-1.05d-11)) then
        tmp = -t
    else if (t <= 62000000.0d0) 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 <= -1.05e-11) {
		tmp = -t;
	} else if (t <= 62000000.0) {
		tmp = -y * z;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.05e-11:
		tmp = -t
	elif t <= 62000000.0:
		tmp = -y * z
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.05e-11)
		tmp = Float64(-t);
	elseif (t <= 62000000.0)
		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 <= -1.05e-11)
		tmp = -t;
	elseif (t <= 62000000.0)
		tmp = -y * z;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.05e-11], (-t), If[LessEqual[t, 62000000.0], N[((-y) * z), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -1.0499999999999999e-11 or 6.2e7 < t

   1. Initial program 91.8%

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

   3. Taylor expanded in 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 68.0

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

   5. Applied rewrites 68.0%

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

   if -1.0499999999999999e-11 < t < 6.2e7

   1. Initial program 78.9%

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

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-log1p.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 24.5

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

   8. Applied rewrites 24.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 24.4%

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

Alternative 14: 46.3% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-log.f64 N/A

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

   6. mul-1-neg N/A

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

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

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. distribute-neg-in N/A

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

   11. neg-mul-1 N/A

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

   12. metadata-eval N/A

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

   13. distribute-rgt-in N/A

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

   14. *-lft-identity N/A

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

   15. lower-fma.f64 N/A

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

   16. neg-mul-1 N/A

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

   17. lower-neg.f64 99.4

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 48.4%

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

Alternative 15: 46.1% accurate, 20.5× speedup

Math

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

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(Float64(-z) * 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 84.8%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-log.f64 N/A

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

   6. mul-1-neg N/A

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

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

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. distribute-neg-in N/A

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

   11. neg-mul-1 N/A

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

   12. metadata-eval N/A

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

   13. distribute-rgt-in N/A

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

   14. *-lft-identity N/A

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

   15. lower-fma.f64 N/A

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

   16. neg-mul-1 N/A

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

   17. lower-neg.f64 99.4

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 48.3%

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

Alternative 16: 35.3% 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 84.8%

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

3. Taylor expanded in 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 32.8

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

5. Applied rewrites 32.8%

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

Reproduce

herbie shell --seed 2024298 
(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))