Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B

Percentage Accurate: 84.9% → 99.8%

Time: 12.7s

Alternatives: 12

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma (log y) x (fma (log1p (- y)) z (- t))))

C

double code(double x, double y, double z, double t) {
	return fma(log(y), x, fma(log1p(-y), z, -t));
}

Julia

function code(x, y, z, t)
	return fma(log(y), x, fma(log1p(Float64(-y)), z, Float64(-t)))
end

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-log.f64 N/A

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

   12. lift--.f64 N/A

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

   13. sub-neg N/A

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

   14. lower-log1p.f64 N/A

       \[\leadsto \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(y\right)\right)}, z, \mathsf{neg}\left(t\right)\right)\right) \]

   15. lower-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\mathsf{log1p}\left(\color{blue}{-y}\right), z, \mathsf{neg}\left(t\right)\right)\right) \]

   16. lower-neg.f64 99.9

       \[\leadsto \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \color{blue}{-t}\right)\right) \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)\right)} \]
5. Add Preprocessing

Alternative 2: 81.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \left(\log \left(1 - y\right) \cdot z + t\_1\right) - t\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-121}:\\ \;\;\;\;t\_1 - t\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+18}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- (+ (* (log (- 1.0 y)) z) t_1) t)))
   (if (<= t_2 -2e-121)
     (- t_1 t)
     (if (<= t_2 5e+18) (* (- y) z) (fma (log y) x (- t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = ((log((1.0 - y)) * z) + t_1) - t;
	double tmp;
	if (t_2 <= -2e-121) {
		tmp = t_1 - t;
	} else if (t_2 <= 5e+18) {
		tmp = -y * z;
	} else {
		tmp = fma(log(y), x, -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(Float64(Float64(log(Float64(1.0 - y)) * z) + t_1) - t)
	tmp = 0.0
	if (t_2 <= -2e-121)
		tmp = Float64(t_1 - t);
	elseif (t_2 <= 5e+18)
		tmp = Float64(Float64(-y) * z);
	else
		tmp = fma(log(y), x, Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision] * z), $MachinePrecision] + t$95$1), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-121], N[(t$95$1 - t), $MachinePrecision], If[LessEqual[t$95$2, 5e+18], N[((-y) * z), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x + (-t)), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 (*.f64 x (log.f64 y)) (*.f64 z (log.f64 (-.f64 #s(literal 1 binary64) y)))) t) < -2e-121

   1. Initial program 88.8%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x \cdot \log y} - t \]
   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 88.1

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

   5. Applied rewrites 88.1%

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

   if -2e-121 < (-.f64 (+.f64 (*.f64 x (log.f64 y)) (*.f64 z (log.f64 (-.f64 #s(literal 1 binary64) y)))) t) < 5e18

   1. Initial program 35.1%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. lower-log1p.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(y\right)\right)}, z, \mathsf{neg}\left(t\right)\right)\right) \]

      15. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\mathsf{log1p}\left(\color{blue}{-y}\right), z, \mathsf{neg}\left(t\right)\right)\right) \]

      16. lower-neg.f64 99.9

          \[\leadsto \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \color{blue}{-t}\right)\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\log y \cdot x + \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\log y \cdot x} + \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right) \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right) + \log y \cdot x} \]

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+r+ N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 N/A

          \[\leadsto \mathsf{log1p}\left(-y\right) \cdot z + \color{blue}{\mathsf{fma}\left(\log y, x, -t\right)} \]

      11. lift-fma.f64 100.0

          \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)} \]

      12. remove-double-div N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}}} \]

      13. lift-/.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}}} \]

      14. inv-pow N/A

          \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}\right)}^{-1}} \]

      15. sqr-pow N/A

          \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}\right)}^{\left(\frac{-1}{2}\right)}} \]

      16. pow2 N/A

          \[\leadsto \color{blue}{{\left({\left(\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}\right)}^{\left(\frac{-1}{2}\right)}\right)}^{2}} \]

      17. lower-pow.f64 N/A

          \[\leadsto \color{blue}{{\left({\left(\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}\right)}^{\left(\frac{-1}{2}\right)}\right)}^{2}} \]

   6. Applied rewrites 60.4%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(z, \mathsf{log1p}\left(-y\right), \mathsf{fma}\left(x, \log y, -t\right)\right)\right)}^{0.5}\right)}^{2}} \]

   7. Taylor expanded in z around inf

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

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

      5. lower-neg.f64 67.5

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

   9. Applied rewrites 67.5%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 67.5%

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

   if 5e18 < (-.f64 (+.f64 (*.f64 x (log.f64 y)) (*.f64 z (log.f64 (-.f64 #s(literal 1 binary64) y)))) t)

   1. Initial program 92.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. lower-log1p.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(y\right)\right)}, z, \mathsf{neg}\left(t\right)\right)\right) \]

      15. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\mathsf{log1p}\left(\color{blue}{-y}\right), z, \mathsf{neg}\left(t\right)\right)\right) \]

      16. lower-neg.f64 99.8

          \[\leadsto \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \color{blue}{-t}\right)\right) \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)\right)} \]

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 92.2

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

   7. Applied rewrites 92.2%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\log \left(1 - y\right) \cdot z + x \cdot \log y\right) - t \leq -2 \cdot 10^{-121}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;\left(\log \left(1 - y\right) \cdot z + x \cdot \log y\right) - t \leq 5 \cdot 10^{+18}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t\_1 - t\\ t_3 := \left(\log \left(1 - y\right) \cdot z + t\_1\right) - t\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-121}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+18}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y)))
        (t_2 (- t_1 t))
        (t_3 (- (+ (* (log (- 1.0 y)) z) t_1) t)))
   (if (<= t_3 -2e-121) t_2 (if (<= t_3 5e+18) (* (- y) z) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = t_1 - t;
	double t_3 = ((log((1.0 - y)) * z) + t_1) - t;
	double tmp;
	if (t_3 <= -2e-121) {
		tmp = t_2;
	} else if (t_3 <= 5e+18) {
		tmp = -y * z;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = x * log(y)
    t_2 = t_1 - t
    t_3 = ((log((1.0d0 - y)) * z) + t_1) - t
    if (t_3 <= (-2d-121)) then
        tmp = t_2
    else if (t_3 <= 5d+18) then
        tmp = -y * z
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double t_2 = t_1 - t;
	double t_3 = ((Math.log((1.0 - y)) * z) + t_1) - t;
	double tmp;
	if (t_3 <= -2e-121) {
		tmp = t_2;
	} else if (t_3 <= 5e+18) {
		tmp = -y * z;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = t_1 - t
	t_3 = ((math.log((1.0 - y)) * z) + t_1) - t
	tmp = 0
	if t_3 <= -2e-121:
		tmp = t_2
	elif t_3 <= 5e+18:
		tmp = -y * z
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(t_1 - t)
	t_3 = Float64(Float64(Float64(log(Float64(1.0 - y)) * z) + t_1) - t)
	tmp = 0.0
	if (t_3 <= -2e-121)
		tmp = t_2;
	elseif (t_3 <= 5e+18)
		tmp = Float64(Float64(-y) * z);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	t_2 = t_1 - t;
	t_3 = ((log((1.0 - y)) * z) + t_1) - t;
	tmp = 0.0;
	if (t_3 <= -2e-121)
		tmp = t_2;
	elseif (t_3 <= 5e+18)
		tmp = -y * z;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - t), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision] * z), $MachinePrecision] + t$95$1), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$3, -2e-121], t$95$2, If[LessEqual[t$95$3, 5e+18], N[((-y) * z), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (*.f64 x (log.f64 y)) (*.f64 z (log.f64 (-.f64 #s(literal 1 binary64) y)))) t) < -2e-121 or 5e18 < (-.f64 (+.f64 (*.f64 x (log.f64 y)) (*.f64 z (log.f64 (-.f64 #s(literal 1 binary64) y)))) t)

   1. Initial program 90.5%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x \cdot \log y} - t \]
   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 90.0

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

   5. Applied rewrites 90.0%

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

   if -2e-121 < (-.f64 (+.f64 (*.f64 x (log.f64 y)) (*.f64 z (log.f64 (-.f64 #s(literal 1 binary64) y)))) t) < 5e18

   1. Initial program 35.1%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. lower-log1p.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(y\right)\right)}, z, \mathsf{neg}\left(t\right)\right)\right) \]

      15. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\mathsf{log1p}\left(\color{blue}{-y}\right), z, \mathsf{neg}\left(t\right)\right)\right) \]

      16. lower-neg.f64 99.9

          \[\leadsto \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \color{blue}{-t}\right)\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\log y \cdot x + \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\log y \cdot x} + \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right) \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right) + \log y \cdot x} \]

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+r+ N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 N/A

          \[\leadsto \mathsf{log1p}\left(-y\right) \cdot z + \color{blue}{\mathsf{fma}\left(\log y, x, -t\right)} \]

      11. lift-fma.f64 100.0

          \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)} \]

      12. remove-double-div N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}}} \]

      13. lift-/.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}}} \]

      14. inv-pow N/A

          \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}\right)}^{-1}} \]

      15. sqr-pow N/A

          \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}\right)}^{\left(\frac{-1}{2}\right)}} \]

      16. pow2 N/A

          \[\leadsto \color{blue}{{\left({\left(\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}\right)}^{\left(\frac{-1}{2}\right)}\right)}^{2}} \]

      17. lower-pow.f64 N/A

          \[\leadsto \color{blue}{{\left({\left(\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}\right)}^{\left(\frac{-1}{2}\right)}\right)}^{2}} \]

   6. Applied rewrites 60.4%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(z, \mathsf{log1p}\left(-y\right), \mathsf{fma}\left(x, \log y, -t\right)\right)\right)}^{0.5}\right)}^{2}} \]

   7. Taylor expanded in z around inf

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

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

      5. lower-neg.f64 67.5

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

   9. Applied rewrites 67.5%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 67.5%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\log \left(1 - y\right) \cdot z + x \cdot \log y\right) - t \leq -2 \cdot 10^{-121}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;\left(\log \left(1 - y\right) \cdot z + x \cdot \log y\right) - t \leq 5 \cdot 10^{+18}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-85}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -3.4e+16)
   (fma (log y) x (- t))
   (if (<= x 5.5e-85) (fma (log1p (- y)) z (- t)) (- (* x (log y)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3.4e+16) {
		tmp = fma(log(y), x, -t);
	} else if (x <= 5.5e-85) {
		tmp = fma(log1p(-y), z, -t);
	} else {
		tmp = (x * log(y)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -3.4e+16)
		tmp = fma(log(y), x, Float64(-t));
	elseif (x <= 5.5e-85)
		tmp = fma(log1p(Float64(-y)), z, Float64(-t));
	else
		tmp = Float64(Float64(x * log(y)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -3.4e+16], N[(N[Log[y], $MachinePrecision] * x + (-t)), $MachinePrecision], If[LessEqual[x, 5.5e-85], N[(N[Log[1 + (-y)], $MachinePrecision] * z + (-t)), $MachinePrecision], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.4e16

   1. Initial program 93.1%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. lower-log1p.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(y\right)\right)}, z, \mathsf{neg}\left(t\right)\right)\right) \]

      15. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\mathsf{log1p}\left(\color{blue}{-y}\right), z, \mathsf{neg}\left(t\right)\right)\right) \]

      16. lower-neg.f64 99.7

          \[\leadsto \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \color{blue}{-t}\right)\right) \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)\right)} \]

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 91.9

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

   7. Applied rewrites 91.9%

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

   if -3.4e16 < x < 5.4999999999999997e-85

   1. Initial program 69.4%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-log1p.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-neg.f64 94.0

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

   5. Applied rewrites 94.0%

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

   if 5.4999999999999997e-85 < x

   1. Initial program 90.7%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x \cdot \log y} - t \]
   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 90.7

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

   5. Applied rewrites 90.7%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-85}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -1.48 \cdot 10^{+138}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 8.1 \cdot 10^{+18}:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -1.48e+138) t_1 (if (<= x 8.1e+18) (- t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -1.48e+138) {
		tmp = t_1;
	} else if (x <= 8.1e+18) {
		tmp = -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 <= (-1.48d+138)) then
        tmp = t_1
    else if (x <= 8.1d+18) then
        tmp = -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 <= -1.48e+138) {
		tmp = t_1;
	} else if (x <= 8.1e+18) {
		tmp = -t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -1.48e+138:
		tmp = t_1
	elif x <= 8.1e+18:
		tmp = -t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -1.48e+138)
		tmp = t_1;
	elseif (x <= 8.1e+18)
		tmp = Float64(-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 <= -1.48e+138)
		tmp = t_1;
	elseif (x <= 8.1e+18)
		tmp = -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[x, -1.48e+138], t$95$1, If[LessEqual[x, 8.1e+18], (-t), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.48000000000000008e138 or 8.1e18 < x

   1. Initial program 93.8%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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 76.3

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

   5. Applied rewrites 76.3%

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

   if -1.48000000000000008e138 < x < 8.1e18

   1. Initial program 74.1%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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 61.4

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

   5. Applied rewrites 61.4%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.48 \cdot 10^{+138}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq 8.1 \cdot 10^{+18}:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-log.f64 N/A

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

   12. lift--.f64 N/A

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

   13. sub-neg N/A

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

   14. lower-log1p.f64 N/A

       \[\leadsto \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(y\right)\right)}, z, \mathsf{neg}\left(t\right)\right)\right) \]

   15. lower-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\mathsf{log1p}\left(\color{blue}{-y}\right), z, \mathsf{neg}\left(t\right)\right)\right) \]

   16. lower-neg.f64 99.9

       \[\leadsto \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \color{blue}{-t}\right)\right) \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)\right)} \]

5. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 99.5

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

7. Applied rewrites 99.5%

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

Alternative 7: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. associate--l- N/A

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

   5. lower--.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-log.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Final simplification 99.5%

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

Alternative 8: 49.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-41}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-37}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, y, -0.5\right), y, -1\right) \cdot y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -6e-41)
   (- t)
   (if (<= t 2.5e-37)
     (* (* (fma (fma -0.3333333333333333 y -0.5) y -1.0) y) z)
     (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -6e-41) {
		tmp = -t;
	} else if (t <= 2.5e-37) {
		tmp = (fma(fma(-0.3333333333333333, y, -0.5), y, -1.0) * y) * z;
	} else {
		tmp = -t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -6e-41)
		tmp = Float64(-t);
	elseif (t <= 2.5e-37)
		tmp = Float64(Float64(fma(fma(-0.3333333333333333, y, -0.5), y, -1.0) * y) * z);
	else
		tmp = Float64(-t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -6e-41], (-t), If[LessEqual[t, 2.5e-37], N[(N[(N[(N[(-0.3333333333333333 * y + -0.5), $MachinePrecision] * y + -1.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -5.99999999999999978e-41 or 2.4999999999999999e-37 < t

   1. Initial program 93.5%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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 63.6

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

   5. Applied rewrites 63.6%

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

   if -5.99999999999999978e-41 < t < 2.4999999999999999e-37

   1. Initial program 63.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. lower-log1p.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(y\right)\right)}, z, \mathsf{neg}\left(t\right)\right)\right) \]

      15. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\mathsf{log1p}\left(\color{blue}{-y}\right), z, \mathsf{neg}\left(t\right)\right)\right) \]

      16. lower-neg.f64 99.8

          \[\leadsto \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \color{blue}{-t}\right)\right) \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\log y \cdot x + \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\log y \cdot x} + \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right) \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right) + \log y \cdot x} \]

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+r+ N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 N/A

          \[\leadsto \mathsf{log1p}\left(-y\right) \cdot z + \color{blue}{\mathsf{fma}\left(\log y, x, -t\right)} \]

      11. lift-fma.f64 99.8

          \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)} \]

      12. remove-double-div N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}}} \]

      13. lift-/.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}}} \]

      14. inv-pow N/A

          \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}\right)}^{-1}} \]

      15. sqr-pow N/A

          \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}\right)}^{\left(\frac{-1}{2}\right)}} \]

      16. pow2 N/A

          \[\leadsto \color{blue}{{\left({\left(\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}\right)}^{\left(\frac{-1}{2}\right)}\right)}^{2}} \]

      17. lower-pow.f64 N/A

          \[\leadsto \color{blue}{{\left({\left(\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}\right)}^{\left(\frac{-1}{2}\right)}\right)}^{2}} \]

   6. Applied rewrites 44.2%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(z, \mathsf{log1p}\left(-y\right), \mathsf{fma}\left(x, \log y, -t\right)\right)\right)}^{0.5}\right)}^{2}} \]

   7. Taylor expanded in z around inf

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

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

      5. lower-neg.f64 38.1

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

   9. Applied rewrites 38.1%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 38.1%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, y, -0.5\right), y, -1\right) \cdot y\right) \cdot z \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 49.3% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-41}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-37}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, y, -1\right) \cdot y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -6e-41)
   (- t)
   (if (<= t 2.5e-37) (* (* (fma -0.5 y -1.0) y) z) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -6e-41) {
		tmp = -t;
	} else if (t <= 2.5e-37) {
		tmp = (fma(-0.5, y, -1.0) * y) * z;
	} else {
		tmp = -t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -6e-41)
		tmp = Float64(-t);
	elseif (t <= 2.5e-37)
		tmp = Float64(Float64(fma(-0.5, y, -1.0) * y) * z);
	else
		tmp = Float64(-t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -6e-41], (-t), If[LessEqual[t, 2.5e-37], N[(N[(N[(-0.5 * y + -1.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -5.99999999999999978e-41 or 2.4999999999999999e-37 < t

   1. Initial program 93.5%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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 63.6

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

   5. Applied rewrites 63.6%

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

   if -5.99999999999999978e-41 < t < 2.4999999999999999e-37

   1. Initial program 63.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. lower-log1p.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(y\right)\right)}, z, \mathsf{neg}\left(t\right)\right)\right) \]

      15. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\mathsf{log1p}\left(\color{blue}{-y}\right), z, \mathsf{neg}\left(t\right)\right)\right) \]

      16. lower-neg.f64 99.8

          \[\leadsto \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \color{blue}{-t}\right)\right) \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\log y \cdot x + \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\log y \cdot x} + \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right) \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right) + \log y \cdot x} \]

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+r+ N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 N/A

          \[\leadsto \mathsf{log1p}\left(-y\right) \cdot z + \color{blue}{\mathsf{fma}\left(\log y, x, -t\right)} \]

      11. lift-fma.f64 99.8

          \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)} \]

      12. remove-double-div N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}}} \]

      13. lift-/.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}}} \]

      14. inv-pow N/A

          \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}\right)}^{-1}} \]

      15. sqr-pow N/A

          \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}\right)}^{\left(\frac{-1}{2}\right)}} \]

      16. pow2 N/A

          \[\leadsto \color{blue}{{\left({\left(\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}\right)}^{\left(\frac{-1}{2}\right)}\right)}^{2}} \]

      17. lower-pow.f64 N/A

          \[\leadsto \color{blue}{{\left({\left(\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}\right)}^{\left(\frac{-1}{2}\right)}\right)}^{2}} \]

   6. Applied rewrites 44.2%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(z, \mathsf{log1p}\left(-y\right), \mathsf{fma}\left(x, \log y, -t\right)\right)\right)}^{0.5}\right)}^{2}} \]

   7. Taylor expanded in z around inf

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

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

      5. lower-neg.f64 38.1

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

   9. Applied rewrites 38.1%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 37.9%

       \[\leadsto \left(\mathsf{fma}\left(-0.5, y, -1\right) \cdot y\right) \cdot z \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 49.1% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-41}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-37}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -6e-41) (- t) (if (<= t 2.5e-37) (* (- y) z) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -6e-41) {
		tmp = -t;
	} else if (t <= 2.5e-37) {
		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 <= (-6d-41)) then
        tmp = -t
    else if (t <= 2.5d-37) 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 <= -6e-41) {
		tmp = -t;
	} else if (t <= 2.5e-37) {
		tmp = -y * z;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -6e-41:
		tmp = -t
	elif t <= 2.5e-37:
		tmp = -y * z
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -6e-41)
		tmp = Float64(-t);
	elseif (t <= 2.5e-37)
		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 <= -6e-41)
		tmp = -t;
	elseif (t <= 2.5e-37)
		tmp = -y * z;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -6e-41], (-t), If[LessEqual[t, 2.5e-37], N[((-y) * z), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -5.99999999999999978e-41 or 2.4999999999999999e-37 < t

   1. Initial program 93.5%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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 63.6

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

   5. Applied rewrites 63.6%

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

   if -5.99999999999999978e-41 < t < 2.4999999999999999e-37

   1. Initial program 63.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. lower-log1p.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(y\right)\right)}, z, \mathsf{neg}\left(t\right)\right)\right) \]

      15. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\mathsf{log1p}\left(\color{blue}{-y}\right), z, \mathsf{neg}\left(t\right)\right)\right) \]

      16. lower-neg.f64 99.8

          \[\leadsto \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \color{blue}{-t}\right)\right) \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\log y \cdot x + \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\log y \cdot x} + \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right) \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right) + \log y \cdot x} \]

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+r+ N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 N/A

          \[\leadsto \mathsf{log1p}\left(-y\right) \cdot z + \color{blue}{\mathsf{fma}\left(\log y, x, -t\right)} \]

      11. lift-fma.f64 99.8

          \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)} \]

      12. remove-double-div N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}}} \]

      13. lift-/.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}}} \]

      14. inv-pow N/A

          \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}\right)}^{-1}} \]

      15. sqr-pow N/A

          \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}\right)}^{\left(\frac{-1}{2}\right)}} \]

      16. pow2 N/A

          \[\leadsto \color{blue}{{\left({\left(\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}\right)}^{\left(\frac{-1}{2}\right)}\right)}^{2}} \]

      17. lower-pow.f64 N/A

          \[\leadsto \color{blue}{{\left({\left(\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}\right)}^{\left(\frac{-1}{2}\right)}\right)}^{2}} \]

   6. Applied rewrites 44.2%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(z, \mathsf{log1p}\left(-y\right), \mathsf{fma}\left(x, \log y, -t\right)\right)\right)}^{0.5}\right)}^{2}} \]

   7. Taylor expanded in z around inf

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

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

      5. lower-neg.f64 38.1

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

   9. Applied rewrites 38.1%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 37.6%

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

Alternative 11: 42.8% accurate, 73.3× 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 81.7%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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 43.9

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

5. Applied rewrites 43.9%

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

Alternative 12: 2.3% accurate, 220.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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 43.9

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

5. Applied rewrites 43.9%

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

7. Applied rewrites 21.8%

   \[\leadsto \frac{0 - t \cdot t}{\color{blue}{0 + t}} \]
8. Step-by-step derivation

9. Applied rewrites 2.2%

   \[\leadsto t \]
10. Add Preprocessing

Developer Target 1: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (-z * (((0.5d0 * (y * y)) + y) + ((0.3333333333333333d0 / (1.0d0 * (1.0d0 * 1.0d0))) * (y * (y * y))))) - (t - (x * log(y)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (-z * (((0.5 * (y * y)) + y) + ((0.3333333333333333 / (1.0 * (1.0 * 1.0))) * (y * (y * y))))) - (t - (x * Math.log(y)));
}

Python

def code(x, y, z, t):
	return (-z * (((0.5 * (y * y)) + y) + ((0.3333333333333333 / (1.0 * (1.0 * 1.0))) * (y * (y * y))))) - (t - (x * math.log(y)))

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = (-z * (((0.5 * (y * y)) + y) + ((0.3333333333333333 / (1.0 * (1.0 * 1.0))) * (y * (y * y))))) - (t - (x * log(y)));
end

Wolfram

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

Reproduce

herbie shell --seed 2024255 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* (- z) (+ (+ (* 1/2 (* y y)) y) (* (/ 1/3 (* 1 (* 1 1))) (* y (* y y))))) (- t (* x (log y)))))

  (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))