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

Percentage Accurate: 85.5% → 99.6%

Time: 16.9s

Alternatives: 15

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: 85.5% 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: 99.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \log y + y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), z \cdot -0.5\right), 0 - z\right)\right) - t \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((x * log(y)) + (y * fma(y, fma(z, (y * fma(y, -0.25, -0.3333333333333333)), (z * -0.5)), (0.0 - z)))) - t;
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * log(y)) + Float64(y * fma(y, fma(z, Float64(y * fma(y, -0.25, -0.3333333333333333)), Float64(z * -0.5)), Float64(0.0 - z)))) - t)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(y * N[(y * N[(z * N[(y * N[(y * -0.25 + -0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(z * -0.5), $MachinePrecision]), $MachinePrecision] + N[(0.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 82.9%

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

3. Taylor expanded in y around 0

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

   1. *-lowering-*.f64 N/A

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

   2. +-commutative N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

5. Simplified 99.8%

   \[\leadsto \left(x \cdot \log y + \color{blue}{y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), z \cdot -0.5\right), 0 - z\right)}\right) - t \]
6. Add Preprocessing

Alternative 2: 76.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t\_1 + z \cdot \log \left(1 - y\right)\\ \mathbf{if}\;t\_2 \leq -8 \cdot 10^{+77}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+123}:\\ \;\;\;\;\mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(y, -0.5, -1\right), 0 - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (+ t_1 (* z (log (- 1.0 y))))))
   (if (<= t_2 -8e+77)
     t_1
     (if (<= t_2 5e+123) (fma z (* y (fma y -0.5 -1.0)) (- 0.0 t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = t_1 + (z * log((1.0 - y)));
	double tmp;
	if (t_2 <= -8e+77) {
		tmp = t_1;
	} else if (t_2 <= 5e+123) {
		tmp = fma(z, (y * fma(y, -0.5, -1.0)), (0.0 - t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(t_1 + Float64(z * log(Float64(1.0 - y))))
	tmp = 0.0
	if (t_2 <= -8e+77)
		tmp = t_1;
	elseif (t_2 <= 5e+123)
		tmp = fma(z, Float64(y * fma(y, -0.5, -1.0)), Float64(0.0 - t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[(z * N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -8e+77], t$95$1, If[LessEqual[t$95$2, 5e+123], N[(z * N[(y * N[(y * -0.5 + -1.0), $MachinePrecision]), $MachinePrecision] + N[(0.0 - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x (log.f64 y)) (*.f64 z (log.f64 (-.f64 #s(literal 1 binary64) y)))) < -7.99999999999999986e77 or 4.99999999999999974e123 < (+.f64 (*.f64 x (log.f64 y)) (*.f64 z (log.f64 (-.f64 #s(literal 1 binary64) y))))

   1. Initial program 94.2%

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

   3. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 99.6%

      \[\leadsto \left(x \cdot \log y + \color{blue}{y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), z \cdot -0.5\right), 0 - z\right)}\right) - t \]

   6. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

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

      2. log-lowering-log.f64 74.0

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

   8. Simplified 74.0%

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

   if -7.99999999999999986e77 < (+.f64 (*.f64 x (log.f64 y)) (*.f64 z (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 4.99999999999999974e123

   1. Initial program 77.6%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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. accelerator-lowering-fma.f64 N/A

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

      3. log-lowering-log.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. neg-sub0 N/A

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

      6. --lowering--.f64 57.3

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

   5. Simplified 57.3%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{y \cdot \left(\frac{-1}{2} \cdot y - 1\right)}, 0 - t\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{y \cdot \left(\frac{-1}{2} \cdot y - 1\right)}, 0 - t\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, y \cdot \color{blue}{\left(\frac{-1}{2} \cdot y + \left(\mathsf{neg}\left(1\right)\right)\right)}, 0 - t\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, y \cdot \left(\color{blue}{y \cdot \frac{-1}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right), 0 - t\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, y \cdot \left(y \cdot \frac{-1}{2} + \color{blue}{-1}\right), 0 - t\right) \]

      5. accelerator-lowering-fma.f64 78.3

         \[\leadsto \mathsf{fma}\left(z, y \cdot \color{blue}{\mathsf{fma}\left(y, -0.5, -1\right)}, 0 - t\right) \]

   8. Simplified 78.3%

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{y \cdot \mathsf{fma}\left(y, -0.5, -1\right)}, 0 - t\right) \]
   9. Step-by-step derivation

      1. neg-sub0 N/A

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

      2. neg-lowering-neg.f64 78.3

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

   10. Applied egg-rr 78.3%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log y + z \cdot \log \left(1 - y\right) \leq -8 \cdot 10^{+77}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \cdot \log y + z \cdot \log \left(1 - y\right) \leq 5 \cdot 10^{+123}:\\ \;\;\;\;\mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(y, -0.5, -1\right), 0 - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

3. Taylor expanded in y around 0

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

   1. *-lowering-*.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. accelerator-lowering-fma.f64 99.8

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

5. Simplified 99.8%

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

Alternative 4: 99.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), 0 - z\right), \mathsf{fma}\left(x, \log y, 0 - t\right)\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma
  y
  (fma z (* y (fma y -0.3333333333333333 -0.5)) (- 0.0 z))
  (fma x (log y) (- 0.0 t))))

C

double code(double x, double y, double z, double t) {
	return fma(y, fma(z, (y * fma(y, -0.3333333333333333, -0.5)), (0.0 - z)), fma(x, log(y), (0.0 - t)));
}

Julia

function code(x, y, z, t)
	return fma(y, fma(z, Float64(y * fma(y, -0.3333333333333333, -0.5)), Float64(0.0 - z)), fma(x, log(y), Float64(0.0 - t)))
end

Wolfram

code[x_, y_, z_, t_] := N[(y * N[(z * N[(y * N[(y * -0.3333333333333333 + -0.5), $MachinePrecision]), $MachinePrecision] + N[(0.0 - z), $MachinePrecision]), $MachinePrecision] + N[(x * N[Log[y], $MachinePrecision] + N[(0.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.9%

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

3. Taylor expanded in y around 0

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

5. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), 0 - z\right), \mathsf{fma}\left(x, \log y, 0 - t\right)\right)} \]
6. Add Preprocessing

Alternative 5: 99.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

3. Taylor expanded in y around 0

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

   1. *-lowering-*.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. accelerator-lowering-fma.f64 99.7

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

5. Simplified 99.7%

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

Alternative 6: 90.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-121}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, 0 - y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, 0 - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -3.2e-121)
   (- (* x (log y)) t)
   (if (<= t 2e-61)
     (fma x (log y) (- 0.0 (* y z)))
     (fma x (log y) (- 0.0 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3.2e-121) {
		tmp = (x * log(y)) - t;
	} else if (t <= 2e-61) {
		tmp = fma(x, log(y), (0.0 - (y * z)));
	} else {
		tmp = fma(x, log(y), (0.0 - t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -3.2e-121)
		tmp = Float64(Float64(x * log(y)) - t);
	elseif (t <= 2e-61)
		tmp = fma(x, log(y), Float64(0.0 - Float64(y * z)));
	else
		tmp = fma(x, log(y), Float64(0.0 - t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -3.2e-121], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t, 2e-61], N[(x * N[Log[y], $MachinePrecision] + N[(0.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Log[y], $MachinePrecision] + N[(0.0 - t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.20000000000000019e-121

   1. Initial program 86.5%

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

   3. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 99.9%

      \[\leadsto \left(x \cdot \log y + \color{blue}{y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), z \cdot -0.5\right), 0 - z\right)}\right) - t \]

   6. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

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

      2. log-lowering-log.f64 86.5

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

   8. Simplified 86.5%

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

   if -3.20000000000000019e-121 < t < 2.0000000000000001e-61

   1. Initial program 71.5%

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

   3. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 99.7%

      \[\leadsto \left(x \cdot \log y + \color{blue}{y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), z \cdot -0.5\right), 0 - z\right)}\right) - t \]

   6. Taylor expanded in y around 0

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

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. log-lowering-log.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-out N/A

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

      8. neg-lowering-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. accelerator-lowering-fma.f64 99.0

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

   8. Simplified 99.0%

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

   9. Taylor expanded in z around inf

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

       1. *-lowering-*.f64 90.7

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

   11. Simplified 90.7%

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

   if 2.0000000000000001e-61 < t

   1. Initial program 94.1%

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

   3. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 99.9%

      \[\leadsto \left(x \cdot \log y + \color{blue}{y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), z \cdot -0.5\right), 0 - z\right)}\right) - t \]

   6. Taylor expanded in y around 0

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

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. log-lowering-log.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-out N/A

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

      8. neg-lowering-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. accelerator-lowering-fma.f64 99.3

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

   8. Simplified 99.3%

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

   9. Taylor expanded in z around 0

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

   11. Simplified 93.1%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-121}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, 0 - y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, 0 - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y \cdot z, \mathsf{fma}\left(y, -0.5, -1\right), \mathsf{fma}\left(x, \log y, 0 - t\right)\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma (* y z) (fma y -0.5 -1.0) (fma x (log y) (- 0.0 t))))

C

double code(double x, double y, double z, double t) {
	return fma((y * z), fma(y, -0.5, -1.0), fma(x, log(y), (0.0 - t)));
}

Julia

function code(x, y, z, t)
	return fma(Float64(y * z), fma(y, -0.5, -1.0), fma(x, log(y), Float64(0.0 - t)))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(y * z), $MachinePrecision] * N[(y * -0.5 + -1.0), $MachinePrecision] + N[(x * N[Log[y], $MachinePrecision] + N[(0.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.9%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. associate-*r* N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. *-lowering-*.f64 N/A

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

   12. sub-neg N/A

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

   13. *-commutative N/A

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

   14. metadata-eval N/A

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

   15. accelerator-lowering-fma.f64 N/A

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

   16. sub-neg N/A

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

   17. remove-double-neg N/A

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

   18. mul-1-neg N/A

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

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

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

5. Simplified 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y, \mathsf{fma}\left(y, -0.5, -1\right), \mathsf{fma}\left(x, \log y, 0 - t\right)\right)} \]

6. Final simplification 99.7%

   \[\leadsto \mathsf{fma}\left(y \cdot z, \mathsf{fma}\left(y, -0.5, -1\right), \mathsf{fma}\left(x, \log y, 0 - t\right)\right) \]
7. Add Preprocessing

Alternative 8: 87.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0 - \mathsf{fma}\left(z, y, t\right)\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{+184}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+278}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- 0.0 (fma z y t))))
   (if (<= z -1.1e+184) t_1 (if (<= z 6.2e+278) (- (* x (log y)) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 0.0 - fma(z, y, t);
	double tmp;
	if (z <= -1.1e+184) {
		tmp = t_1;
	} else if (z <= 6.2e+278) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(0.0 - fma(z, y, t))
	tmp = 0.0
	if (z <= -1.1e+184)
		tmp = t_1;
	elseif (z <= 6.2e+278)
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(0.0 - N[(z * y + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.1e+184], t$95$1, If[LessEqual[z, 6.2e+278], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.1e184 or 6.19999999999999943e278 < z

   1. Initial program 35.0%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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. accelerator-lowering-fma.f64 N/A

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

      3. log-lowering-log.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. neg-sub0 N/A

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

      6. --lowering--.f64 25.1

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

   5. Simplified 25.1%

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

   6. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-out N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 87.3

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

   8. Simplified 87.3%

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

   if -1.1e184 < z < 6.19999999999999943e278

   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 y around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 99.8%

      \[\leadsto \left(x \cdot \log y + \color{blue}{y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), z \cdot -0.5\right), 0 - z\right)}\right) - t \]

   6. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

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

      2. log-lowering-log.f64 89.7

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

   8. Simplified 89.7%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+184}:\\ \;\;\;\;0 - \mathsf{fma}\left(z, y, t\right)\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+278}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;0 - \mathsf{fma}\left(z, y, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \log y, 0 - \mathsf{fma}\left(z, y, t\right)\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (fma x (log y) (- 0.0 (fma z y t))))

C

double code(double x, double y, double z, double t) {
	return fma(x, log(y), (0.0 - fma(z, y, t)));
}

Julia

function code(x, y, z, t)
	return fma(x, log(y), Float64(0.0 - fma(z, y, t)))
end

Wolfram

code[x_, y_, z_, t_] := N[(x * N[Log[y], $MachinePrecision] + N[(0.0 - N[(z * y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.9%

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

3. Taylor expanded in y around 0

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

   1. *-lowering-*.f64 N/A

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

   2. +-commutative N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

5. Simplified 99.8%

   \[\leadsto \left(x \cdot \log y + \color{blue}{y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), z \cdot -0.5\right), 0 - z\right)}\right) - t \]

6. Taylor expanded in y around 0

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

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. log-lowering-log.f64 N/A

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

   5. sub-neg N/A

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

   6. mul-1-neg N/A

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

   7. distribute-neg-out N/A

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

   8. neg-lowering-neg.f64 N/A

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

   9. *-commutative N/A

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

   10. accelerator-lowering-fma.f64 99.4

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

8. Simplified 99.4%

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

9. Final simplification 99.4%

   \[\leadsto \mathsf{fma}\left(x, \log y, 0 - \mathsf{fma}\left(z, y, t\right)\right) \]
10. Add Preprocessing

Alternative 10: 57.9% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), -1\right), 0 - t\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma z (* y (fma y (fma y -0.3333333333333333 -0.5) -1.0)) (- 0.0 t)))

C

double code(double x, double y, double z, double t) {
	return fma(z, (y * fma(y, fma(y, -0.3333333333333333, -0.5), -1.0)), (0.0 - t));
}

Julia

function code(x, y, z, t)
	return fma(z, Float64(y * fma(y, fma(y, -0.3333333333333333, -0.5), -1.0)), Float64(0.0 - t))
end

Wolfram

code[x_, y_, z_, t_] := N[(z * N[(y * N[(y * N[(y * -0.3333333333333333 + -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(0.0 - t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.9%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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. accelerator-lowering-fma.f64 N/A

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

   3. log-lowering-log.f64 N/A

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

   4. --lowering--.f64 N/A

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

   5. neg-sub0 N/A

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

   6. --lowering--.f64 45.4

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

5. Simplified 45.4%

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

6. Taylor expanded in y around 0

   \[\leadsto \mathsf{fma}\left(z, \color{blue}{y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)}, 0 - t\right) \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)}, 0 - t\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(z, y \cdot \color{blue}{\left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}, 0 - t\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(z, y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) + \color{blue}{-1}\right), 0 - t\right) \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(z, y \cdot \color{blue}{\mathsf{fma}\left(y, \frac{-1}{3} \cdot y - \frac{1}{2}, -1\right)}, 0 - t\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(y, \color{blue}{\frac{-1}{3} \cdot y + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right), 0 - t\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{-1}{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), -1\right), 0 - t\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{3} + \color{blue}{\frac{-1}{2}}, -1\right), 0 - t\right) \]

   8. accelerator-lowering-fma.f64 61.2

      \[\leadsto \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, -0.3333333333333333, -0.5\right)}, -1\right), 0 - t\right) \]

8. Simplified 61.2%

   \[\leadsto \mathsf{fma}\left(z, \color{blue}{y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), -1\right)}, 0 - t\right) \]
9. Add Preprocessing

Alternative 11: 57.7% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(y, -0.5, -1\right), 0 - t\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma z (* y (fma y -0.5 -1.0)) (- 0.0 t)))

C

double code(double x, double y, double z, double t) {
	return fma(z, (y * fma(y, -0.5, -1.0)), (0.0 - t));
}

Julia

function code(x, y, z, t)
	return fma(z, Float64(y * fma(y, -0.5, -1.0)), Float64(0.0 - t))
end

Wolfram

code[x_, y_, z_, t_] := N[(z * N[(y * N[(y * -0.5 + -1.0), $MachinePrecision]), $MachinePrecision] + N[(0.0 - t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.9%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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. accelerator-lowering-fma.f64 N/A

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

   3. log-lowering-log.f64 N/A

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

   4. --lowering--.f64 N/A

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

   5. neg-sub0 N/A

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

   6. --lowering--.f64 45.4

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

5. Simplified 45.4%

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

6. Taylor expanded in y around 0

   \[\leadsto \mathsf{fma}\left(z, \color{blue}{y \cdot \left(\frac{-1}{2} \cdot y - 1\right)}, 0 - t\right) \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{y \cdot \left(\frac{-1}{2} \cdot y - 1\right)}, 0 - t\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(z, y \cdot \color{blue}{\left(\frac{-1}{2} \cdot y + \left(\mathsf{neg}\left(1\right)\right)\right)}, 0 - t\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(z, y \cdot \left(\color{blue}{y \cdot \frac{-1}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right), 0 - t\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(z, y \cdot \left(y \cdot \frac{-1}{2} + \color{blue}{-1}\right), 0 - t\right) \]

   5. accelerator-lowering-fma.f64 61.1

      \[\leadsto \mathsf{fma}\left(z, y \cdot \color{blue}{\mathsf{fma}\left(y, -0.5, -1\right)}, 0 - t\right) \]

8. Simplified 61.1%

   \[\leadsto \mathsf{fma}\left(z, \color{blue}{y \cdot \mathsf{fma}\left(y, -0.5, -1\right)}, 0 - t\right) \]
9. Step-by-step derivation

   1. neg-sub0 N/A

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

   2. neg-lowering-neg.f64 61.1

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

10. Applied egg-rr 61.1%

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

11. Final simplification 61.1%

    \[\leadsto \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(y, -0.5, -1\right), 0 - t\right) \]
12. Add Preprocessing

Alternative 12: 57.7% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.9%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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. accelerator-lowering-fma.f64 N/A

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

   3. log-lowering-log.f64 N/A

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

   4. --lowering--.f64 N/A

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

   5. neg-sub0 N/A

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

   6. --lowering--.f64 45.4

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

5. Simplified 45.4%

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

6. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. mul-1-neg N/A

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

   5. distribute-rgt-in N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. --lowering--.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. sub-neg N/A

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

   12. *-commutative N/A

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

   13. metadata-eval N/A

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

   14. accelerator-lowering-fma.f64 61.1

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

8. Simplified 61.1%

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

Alternative 13: 57.5% accurate, 22.0× speedup

Math

\[\begin{array}{l} \\ 0 - \mathsf{fma}\left(z, y, t\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- 0.0 (fma z y t)))

C

double code(double x, double y, double z, double t) {
	return 0.0 - fma(z, y, t);
}

Julia

function code(x, y, z, t)
	return Float64(0.0 - fma(z, y, t))
end

Wolfram

code[x_, y_, z_, t_] := N[(0.0 - N[(z * y + t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.9%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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. accelerator-lowering-fma.f64 N/A

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

   3. log-lowering-log.f64 N/A

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

   4. --lowering--.f64 N/A

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

   5. neg-sub0 N/A

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

   6. --lowering--.f64 45.4

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

5. Simplified 45.4%

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

6. Taylor expanded in y around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. distribute-neg-out N/A

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

   4. neg-lowering-neg.f64 N/A

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

   5. *-commutative N/A

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

   6. accelerator-lowering-fma.f64 60.8

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

8. Simplified 60.8%

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

9. Final simplification 60.8%

   \[\leadsto 0 - \mathsf{fma}\left(z, y, t\right) \]
10. Add Preprocessing

Alternative 14: 43.2% accurate, 55.0× speedup

Math

\[\begin{array}{l} \\ 0 - t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- 0.0 t))

C

double code(double x, double y, double z, double t) {
	return 0.0 - 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 = 0.0d0 - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return 0.0 - t;
}

Python

def code(x, y, z, t):
	return 0.0 - t

Julia

function code(x, y, z, t)
	return Float64(0.0 - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 0.0 - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(0.0 - t), $MachinePrecision]

Derivation

1. Initial program 82.9%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - t} \]

   3. --lowering--.f64 44.7

      \[\leadsto \color{blue}{0 - t} \]

5. Simplified 44.7%

   \[\leadsto \color{blue}{0 - t} \]
6. Step-by-step derivation

   1. sub0-neg N/A

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

   2. +-lft-identity N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(0 + t\right)}\right) \]

   3. neg-lowering-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\left(0 + t\right)\right)} \]

   4. +-lft-identity 44.7

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

7. Applied egg-rr 44.7%

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

8. Final simplification 44.7%

   \[\leadsto 0 - t \]
9. Add Preprocessing

Alternative 15: 2.2% 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 82.9%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - t} \]

   3. --lowering--.f64 44.7

      \[\leadsto \color{blue}{0 - t} \]

5. Simplified 44.7%

   \[\leadsto \color{blue}{0 - t} \]
6. Step-by-step derivation

   1. sub0-neg N/A

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

   2. +-lft-identity N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(0 + t\right)}\right) \]

   3. neg-lowering-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\left(0 + t\right)\right)} \]

   4. +-lft-identity 44.7

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

7. Applied egg-rr 44.7%

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

   1. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - t} \]

   2. flip3-- N/A

      \[\leadsto \color{blue}{\frac{{0}^{3} - {t}^{3}}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0} - {t}^{3}}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \]

   4. pow3 N/A

      \[\leadsto \frac{0 - \color{blue}{\left(t \cdot t\right) \cdot t}}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \]

   5. +-rgt-identity N/A

      \[\leadsto \frac{0 - \color{blue}{\left(t \cdot t + 0\right)} \cdot t}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \]

   6. sub0-neg N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(t \cdot t + 0\right) \cdot t\right)}}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \]

   7. +-rgt-identity N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \]

   8. pow3 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{t}^{3}}\right)}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \]

   9. cube-neg N/A

      \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(t\right)\right)}^{3}}}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \]

   10. neg-sub0 N/A

       \[\leadsto \frac{{\color{blue}{\left(0 - t\right)}}^{3}}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \]

   11. sqr-pow N/A

       \[\leadsto \frac{\color{blue}{{\left(0 - t\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(0 - t\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \]

   12. unpow-prod-down N/A

       \[\leadsto \frac{\color{blue}{{\left(\left(0 - t\right) \cdot \left(0 - t\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \]

   13. neg-sub0 N/A

       \[\leadsto \frac{{\left(\color{blue}{\left(\mathsf{neg}\left(t\right)\right)} \cdot \left(0 - t\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \]

   14. neg-sub0 N/A

       \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(t\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \]

   15. sqr-neg N/A

       \[\leadsto \frac{{\color{blue}{\left(t \cdot t\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \]

   16. unpow-prod-down N/A

       \[\leadsto \frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \]

   17. sqr-pow N/A

       \[\leadsto \frac{\color{blue}{{t}^{3}}}{0 \cdot 0 + \left(t \cdot t + 0 \cdot t\right)} \]

   18. metadata-eval N/A

       \[\leadsto \frac{{t}^{3}}{\color{blue}{0} + \left(t \cdot t + 0 \cdot t\right)} \]

   19. +-lft-identity N/A

       \[\leadsto \frac{{t}^{3}}{\color{blue}{t \cdot t + 0 \cdot t}} \]

   20. mul0-lft N/A

       \[\leadsto \frac{{t}^{3}}{t \cdot t + \color{blue}{0}} \]

   21. +-rgt-identity N/A

       \[\leadsto \frac{{t}^{3}}{\color{blue}{t \cdot t}} \]

   22. pow2 N/A

       \[\leadsto \frac{{t}^{3}}{\color{blue}{{t}^{2}}} \]

   23. pow-div N/A

       \[\leadsto \color{blue}{{t}^{\left(3 - 2\right)}} \]

   24. metadata-eval N/A

       \[\leadsto {t}^{\color{blue}{1}} \]

   25. unpow1 2.1

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

9. Applied egg-rr 2.1%

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

Developer Target 1: 99.5% 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 2024195 
(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))