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

Percentage Accurate: 89.2% → 99.6%

Time: 18.5s

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: 89.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x - 1.0d0) * log(y)) + ((z - 1.0d0) * log((1.0d0 - y)))) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

3. Taylor expanded in y around 0

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

   1. lower-fma.f64 N/A

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

5. Applied rewrites 99.5%

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

6. Final simplification 99.5%

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

Alternative 2: 94.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;x + -1 \leq -5 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x + -1 \leq 2 \cdot 10^{+53}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - z, -\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t)))
   (if (<= (+ x -1.0) -5e+21)
     t_1
     (if (<= (+ x -1.0) 2e+53) (- (fma y (- 1.0 z) (- (log y))) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - t;
	double tmp;
	if ((x + -1.0) <= -5e+21) {
		tmp = t_1;
	} else if ((x + -1.0) <= 2e+53) {
		tmp = fma(y, (1.0 - z), -log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - t)
	tmp = 0.0
	if (Float64(x + -1.0) <= -5e+21)
		tmp = t_1;
	elseif (Float64(x + -1.0) <= 2e+53)
		tmp = Float64(fma(y, Float64(1.0 - z), 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 * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[N[(x + -1.0), $MachinePrecision], -5e+21], t$95$1, If[LessEqual[N[(x + -1.0), $MachinePrecision], 2e+53], N[(N[(y * N[(1.0 - z), $MachinePrecision] + (-N[Log[y], $MachinePrecision])), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -5e21 or 2e53 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 95.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 95.3

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

   5. Applied rewrites 95.3%

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

   if -5e21 < (-.f64 x #s(literal 1 binary64)) < 2e53

   1. Initial program 82.7%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.7%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + -1 \leq -5 \cdot 10^{+21}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x + -1 \leq 2 \cdot 10^{+53}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - z, -\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;x + -1 \leq -5 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x + -1 \leq 2 \cdot 10^{+53}:\\ \;\;\;\;\left(\mathsf{fma}\left(y, -z, y\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t)))
   (if (<= (+ x -1.0) -5e+21)
     t_1
     (if (<= (+ x -1.0) 2e+53) (- (- (fma y (- z) y) (log y)) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - t;
	double tmp;
	if ((x + -1.0) <= -5e+21) {
		tmp = t_1;
	} else if ((x + -1.0) <= 2e+53) {
		tmp = (fma(y, -z, y) - log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - t)
	tmp = 0.0
	if (Float64(x + -1.0) <= -5e+21)
		tmp = t_1;
	elseif (Float64(x + -1.0) <= 2e+53)
		tmp = Float64(Float64(fma(y, Float64(-z), y) - log(y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[N[(x + -1.0), $MachinePrecision], -5e+21], t$95$1, If[LessEqual[N[(x + -1.0), $MachinePrecision], 2e+53], N[(N[(N[(y * (-z) + y), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -5e21 or 2e53 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 95.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 95.3

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

   5. Applied rewrites 95.3%

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

   if -5e21 < (-.f64 x #s(literal 1 binary64)) < 2e53

   1. Initial program 82.7%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.7%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + -1 \leq -5 \cdot 10^{+21}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x + -1 \leq 2 \cdot 10^{+53}:\\ \;\;\;\;\left(\mathsf{fma}\left(y, -z, y\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 99.5%

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

8. Final simplification 99.5%

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

Alternative 5: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

5. Applied rewrites 99.5%

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

6. Final simplification 99.5%

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

Alternative 6: 87.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;x + -1 \leq -20000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x + -1 \leq -0.99999999999995:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t)))
   (if (<= (+ x -1.0) -20000000000000.0)
     t_1
     (if (<= (+ x -1.0) -0.99999999999995) (- (- (log y)) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - t;
	double tmp;
	if ((x + -1.0) <= -20000000000000.0) {
		tmp = t_1;
	} else if ((x + -1.0) <= -0.99999999999995) {
		tmp = -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) :: tmp
    t_1 = (x * log(y)) - t
    if ((x + (-1.0d0)) <= (-20000000000000.0d0)) then
        tmp = t_1
    else if ((x + (-1.0d0)) <= (-0.99999999999995d0)) then
        tmp = -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 = (x * Math.log(y)) - t;
	double tmp;
	if ((x + -1.0) <= -20000000000000.0) {
		tmp = t_1;
	} else if ((x + -1.0) <= -0.99999999999995) {
		tmp = -Math.log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - t
	tmp = 0
	if (x + -1.0) <= -20000000000000.0:
		tmp = t_1
	elif (x + -1.0) <= -0.99999999999995:
		tmp = -math.log(y) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - t)
	tmp = 0.0
	if (Float64(x + -1.0) <= -20000000000000.0)
		tmp = t_1;
	elseif (Float64(x + -1.0) <= -0.99999999999995)
		tmp = Float64(Float64(-log(y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * log(y)) - t;
	tmp = 0.0;
	if ((x + -1.0) <= -20000000000000.0)
		tmp = t_1;
	elseif ((x + -1.0) <= -0.99999999999995)
		tmp = -log(y) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[N[(x + -1.0), $MachinePrecision], -20000000000000.0], t$95$1, If[LessEqual[N[(x + -1.0), $MachinePrecision], -0.99999999999995], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -2e13 or -0.99999999999995004 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 93.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 93.4

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

   5. Applied rewrites 93.4%

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

   if -2e13 < (-.f64 x #s(literal 1 binary64)) < -0.99999999999995004

   1. Initial program 83.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 81.6

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

   5. Applied rewrites 81.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 81.5%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + -1 \leq -20000000000000:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x + -1 \leq -0.99999999999995:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x + -1 \leq -4 \cdot 10^{+58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x + -1 \leq 2 \cdot 10^{+61}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= (+ x -1.0) -4e+58)
     t_1
     (if (<= (+ x -1.0) 2e+61) (- (- (log y)) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if ((x + -1.0) <= -4e+58) {
		tmp = t_1;
	} else if ((x + -1.0) <= 2e+61) {
		tmp = -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) :: tmp
    t_1 = x * log(y)
    if ((x + (-1.0d0)) <= (-4d+58)) then
        tmp = t_1
    else if ((x + (-1.0d0)) <= 2d+61) then
        tmp = -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 = x * Math.log(y);
	double tmp;
	if ((x + -1.0) <= -4e+58) {
		tmp = t_1;
	} else if ((x + -1.0) <= 2e+61) {
		tmp = -Math.log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if (x + -1.0) <= -4e+58:
		tmp = t_1
	elif (x + -1.0) <= 2e+61:
		tmp = -math.log(y) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (Float64(x + -1.0) <= -4e+58)
		tmp = t_1;
	elseif (Float64(x + -1.0) <= 2e+61)
		tmp = Float64(Float64(-log(y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if ((x + -1.0) <= -4e+58)
		tmp = t_1;
	elseif ((x + -1.0) <= 2e+61)
		tmp = -log(y) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x + -1.0), $MachinePrecision], -4e+58], t$95$1, If[LessEqual[N[(x + -1.0), $MachinePrecision], 2e+61], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -3.99999999999999978e58 or 1.9999999999999999e61 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 94.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 80.4

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

   5. Applied rewrites 80.4%

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

   if -3.99999999999999978e58 < (-.f64 x #s(literal 1 binary64)) < 1.9999999999999999e61

   1. Initial program 83.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 82.7

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

   5. Applied rewrites 82.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 79.5%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + -1 \leq -4 \cdot 10^{+58}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x + -1 \leq 2 \cdot 10^{+61}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x + -1 \leq -4 \cdot 10^{+58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x + -1 \leq 2 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= (+ x -1.0) -4e+58)
     t_1
     (if (<= (+ x -1.0) 2e+61) (- (fma y (- z) y) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if ((x + -1.0) <= -4e+58) {
		tmp = t_1;
	} else if ((x + -1.0) <= 2e+61) {
		tmp = fma(y, -z, y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (Float64(x + -1.0) <= -4e+58)
		tmp = t_1;
	elseif (Float64(x + -1.0) <= 2e+61)
		tmp = Float64(fma(y, Float64(-z), y) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x + -1.0), $MachinePrecision], -4e+58], t$95$1, If[LessEqual[N[(x + -1.0), $MachinePrecision], 2e+61], N[(N[(y * (-z) + y), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -3.99999999999999978e58 or 1.9999999999999999e61 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 94.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 80.4

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

   5. Applied rewrites 80.4%

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

   if -3.99999999999999978e58 < (-.f64 x #s(literal 1 binary64)) < 1.9999999999999999e61

   1. Initial program 83.9%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 62.4%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + -1 \leq -4 \cdot 10^{+58}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x + -1 \leq 2 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 88.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z + -1 \leq -2 \cdot 10^{+162}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x + -1, y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ z -1.0) -2e+162)
   (- (* z (log1p (- y))) t)
   (- (fma (log y) (+ x -1.0) y) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z + -1.0) <= -2e+162) {
		tmp = (z * log1p(-y)) - t;
	} else {
		tmp = fma(log(y), (x + -1.0), y) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z + -1.0) <= -2e+162)
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	else
		tmp = Float64(fma(log(y), Float64(x + -1.0), y) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z + -1.0), $MachinePrecision], -2e+162], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision] + y), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z #s(literal 1 binary64)) < -1.9999999999999999e162

   1. Initial program 55.4%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-neg.f64 67.4

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

   5. Applied rewrites 67.4%

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

   if -1.9999999999999999e162 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 92.9%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 92.7%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + -1 \leq -2 \cdot 10^{+162}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x + -1, y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 88.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z + -1 \leq -2 \cdot 10^{+162}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(x + -1\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ z -1.0) -2e+162)
   (- (* z (log1p (- y))) t)
   (- (* (log y) (+ x -1.0)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z + -1.0) <= -2e+162) {
		tmp = (z * log1p(-y)) - t;
	} else {
		tmp = (log(y) * (x + -1.0)) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z + -1.0) <= -2e+162) {
		tmp = (z * Math.log1p(-y)) - t;
	} else {
		tmp = (Math.log(y) * (x + -1.0)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z + -1.0) <= -2e+162:
		tmp = (z * math.log1p(-y)) - t
	else:
		tmp = (math.log(y) * (x + -1.0)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z + -1.0) <= -2e+162)
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	else
		tmp = Float64(Float64(log(y) * Float64(x + -1.0)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z + -1.0), $MachinePrecision], -2e+162], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z #s(literal 1 binary64)) < -1.9999999999999999e162

   1. Initial program 55.4%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-neg.f64 67.4

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

   5. Applied rewrites 67.4%

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

   if -1.9999999999999999e162 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 92.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 92.5

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

   5. Applied rewrites 92.5%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + -1 \leq -2 \cdot 10^{+162}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(x + -1\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. associate--l- N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-log.f64 N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. sub-neg N/A

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

   14. metadata-eval N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Final simplification 99.5%

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

Alternative 12: 43.2% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{+19}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 34000000000:\\ \;\;\;\;\mathsf{fma}\left(y, -z, y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -3.8e+19) (- t) (if (<= t 34000000000.0) (fma y (- z) y) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3.8e+19) {
		tmp = -t;
	} else if (t <= 34000000000.0) {
		tmp = fma(y, -z, y);
	} else {
		tmp = -t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -3.8e+19)
		tmp = Float64(-t);
	elseif (t <= 34000000000.0)
		tmp = fma(y, Float64(-z), y);
	else
		tmp = Float64(-t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -3.8e+19], (-t), If[LessEqual[t, 34000000000.0], N[(y * (-z) + y), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -3.8e19 or 3.4e10 < t

   1. Initial program 95.3%

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

   3. Taylor expanded in 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 72.0

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

   5. Applied rewrites 72.0%

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

   if -3.8e19 < t < 3.4e10

   1. Initial program 82.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 99.3%

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

   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. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-out N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. lower-+.f64 99.1

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

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 20.8%

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

Alternative 13: 42.9% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{+19}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 34000000000:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -3.8e+19) (- t) (if (<= t 34000000000.0) (* z (- y)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3.8e+19) {
		tmp = -t;
	} else if (t <= 34000000000.0) {
		tmp = 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 <= (-3.8d+19)) then
        tmp = -t
    else if (t <= 34000000000.0d0) then
        tmp = 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 <= -3.8e+19) {
		tmp = -t;
	} else if (t <= 34000000000.0) {
		tmp = z * -y;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -3.8e+19:
		tmp = -t
	elif t <= 34000000000.0:
		tmp = z * -y
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -3.8e+19)
		tmp = Float64(-t);
	elseif (t <= 34000000000.0)
		tmp = Float64(z * Float64(-y));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -3.8e+19)
		tmp = -t;
	elseif (t <= 34000000000.0)
		tmp = z * -y;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -3.8e+19], (-t), If[LessEqual[t, 34000000000.0], N[(z * (-y)), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -3.8e19 or 3.4e10 < t

   1. Initial program 95.3%

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

   3. Taylor expanded in 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 72.0

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

   5. Applied rewrites 72.0%

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

   if -3.8e19 < t < 3.4e10

   1. Initial program 82.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 99.3%

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

   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. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-out N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. lower-+.f64 99.1

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

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 20.2%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{+19}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 34000000000:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.3% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

3. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

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

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

   3. mul-1-neg N/A

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

   4. lower-fma.f64 N/A

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

   5. mul-1-neg N/A

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

   6. neg-sub0 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

   10. associate--r+ N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-log.f64 N/A

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

   15. sub-neg N/A

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

   16. metadata-eval N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 46.4%

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

Alternative 15: 46.1% accurate, 20.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

3. Taylor expanded in 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 46.5

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

5. Applied rewrites 46.5%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 46.2%

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

9. Final simplification 46.2%

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

Alternative 16: 35.8% 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 88.0%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 34.6

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

5. Applied rewrites 34.6%

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

Reproduce

herbie shell --seed 2024221 
(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))