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

Percentage Accurate: 89.3% → 99.6%

Time: 21.3s

Alternatives: 18

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.5%

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

3. Taylor expanded in y around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

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

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. accelerator-lowering-fma.f64 99.8

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

5. Simplified 99.8%

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. +-lowering-+.f64 N/A

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

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

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

   15. accelerator-lowering-fma.f64 99.8

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 86.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot \left(-1 + x\right)\\ t_2 := \left(t\_1 + \left(-1 + z\right) \cdot \log \left(1 - y\right)\right) - t\\ t_3 := x \cdot \log y - t\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+27}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 10^{+15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+46}:\\ \;\;\;\;y - \mathsf{fma}\left(y, z, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) (+ -1.0 x)))
        (t_2 (- (+ t_1 (* (+ -1.0 z) (log (- 1.0 y)))) t))
        (t_3 (- (* x (log y)) t)))
   (if (<= t_2 -5e+27)
     t_3
     (if (<= t_2 1e+15) t_1 (if (<= t_2 5e+46) (- y (fma y z t)) t_3)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * (-1.0 + x);
	double t_2 = (t_1 + ((-1.0 + z) * log((1.0 - y)))) - t;
	double t_3 = (x * log(y)) - t;
	double tmp;
	if (t_2 <= -5e+27) {
		tmp = t_3;
	} else if (t_2 <= 1e+15) {
		tmp = t_1;
	} else if (t_2 <= 5e+46) {
		tmp = y - fma(y, z, t);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * Float64(-1.0 + x))
	t_2 = Float64(Float64(t_1 + Float64(Float64(-1.0 + z) * log(Float64(1.0 - y)))) - t)
	t_3 = Float64(Float64(x * log(y)) - t)
	tmp = 0.0
	if (t_2 <= -5e+27)
		tmp = t_3;
	elseif (t_2 <= 1e+15)
		tmp = t_1;
	elseif (t_2 <= 5e+46)
		tmp = Float64(y - fma(y, z, t));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 + N[(N[(-1.0 + z), $MachinePrecision] * N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+27], t$95$3, If[LessEqual[t$95$2, 1e+15], t$95$1, If[LessEqual[t$95$2, 5e+46], N[(y - N[(y * z + t), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) t) < -4.99999999999999979e27 or 5.0000000000000002e46 < (-.f64 (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) t)

   1. Initial program 92.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

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

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

      3. log-lowering-log.f64 91.5

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

   5. Simplified 91.5%

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

   if -4.99999999999999979e27 < (-.f64 (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) t) < 1e15

   1. Initial program 79.5%

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

   3. Taylor expanded in y around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 79.4

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

   5. Simplified 79.4%

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

   6. Taylor expanded in t around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 76.4

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

   8. Simplified 76.4%

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

   if 1e15 < (-.f64 (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) t) < 5.0000000000000002e46

   1. Initial program 33.3%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. +-lowering-+.f64 N/A

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

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

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

      15. sub-neg N/A

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

      16. *-commutative N/A

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

      17. metadata-eval N/A

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

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

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

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

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

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

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

      21. sub-neg N/A

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. distribute-lft-out N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. distribute-neg-in N/A

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

      14. mul-1-neg N/A

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

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

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. +-commutative N/A

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

      19. +-lowering-+.f64 N/A

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

      20. mul-1-neg N/A

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

      21. distribute-neg-in N/A

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

      22. metadata-eval N/A

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

      23. distribute-neg-frac N/A

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

      24. metadata-eval N/A

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

      25. +-lowering-+.f64 N/A

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

      26. /-lowering-/.f64 51.3

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

   8. Simplified 51.3%

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

   9. Taylor expanded in y around 0

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

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

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

       3. associate--r+ N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. distribute-lft-in N/A

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

       7. *-commutative N/A

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

       8. associate-+l+ N/A

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

       9. +-commutative N/A

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

       10. +-commutative N/A

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

       11. associate-+r+ N/A

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

       12. +-commutative N/A

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

       13. associate-+l+ N/A

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

       14. associate--r+ N/A

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

       15. neg-sub0 N/A

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

       16. mul-1-neg N/A

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

       17. remove-double-neg N/A

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

       18. --lowering--.f64 N/A

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

       19. accelerator-lowering-fma.f64 80.7

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

   11. Simplified 80.7%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\log y \cdot \left(-1 + x\right) + \left(-1 + z\right) \cdot \log \left(1 - y\right)\right) - t \leq -5 \cdot 10^{+27}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;\left(\log y \cdot \left(-1 + x\right) + \left(-1 + z\right) \cdot \log \left(1 - y\right)\right) - t \leq 10^{+15}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right)\\ \mathbf{elif}\;\left(\log y \cdot \left(-1 + x\right) + \left(-1 + z\right) \cdot \log \left(1 - y\right)\right) - t \leq 5 \cdot 10^{+46}:\\ \;\;\;\;y - \mathsf{fma}\left(y, z, t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;-1 + x \leq -2 \cdot 10^{+121}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;-1 + x \leq -1.000000000010413:\\ \;\;\;\;y - \mathsf{fma}\left(y, z, t\right)\\ \mathbf{elif}\;-1 + x \leq -0.99:\\ \;\;\;\;0 - \left(\log y + t\right)\\ \mathbf{elif}\;-1 + x \leq 10^{+63}:\\ \;\;\;\;z \cdot \left(y \cdot \mathsf{fma}\left(y, -0.5, -1\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= (+ -1.0 x) -2e+121)
     t_1
     (if (<= (+ -1.0 x) -1.000000000010413)
       (- y (fma y z t))
       (if (<= (+ -1.0 x) -0.99)
         (- 0.0 (+ (log y) t))
         (if (<= (+ -1.0 x) 1e+63)
           (- (* z (* y (fma y -0.5 -1.0))) t)
           t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if ((-1.0 + x) <= -2e+121) {
		tmp = t_1;
	} else if ((-1.0 + x) <= -1.000000000010413) {
		tmp = y - fma(y, z, t);
	} else if ((-1.0 + x) <= -0.99) {
		tmp = 0.0 - (log(y) + t);
	} else if ((-1.0 + x) <= 1e+63) {
		tmp = (z * (y * fma(y, -0.5, -1.0))) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (Float64(-1.0 + x) <= -2e+121)
		tmp = t_1;
	elseif (Float64(-1.0 + x) <= -1.000000000010413)
		tmp = Float64(y - fma(y, z, t));
	elseif (Float64(-1.0 + x) <= -0.99)
		tmp = Float64(0.0 - Float64(log(y) + t));
	elseif (Float64(-1.0 + x) <= 1e+63)
		tmp = Float64(Float64(z * Float64(y * fma(y, -0.5, -1.0))) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-1.0 + x), $MachinePrecision], -2e+121], t$95$1, If[LessEqual[N[(-1.0 + x), $MachinePrecision], -1.000000000010413], N[(y - N[(y * z + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-1.0 + x), $MachinePrecision], -0.99], N[(0.0 - N[(N[Log[y], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-1.0 + x), $MachinePrecision], 1e+63], N[(N[(z * N[(y * N[(y * -0.5 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -2.00000000000000007e121 or 1.00000000000000006e63 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 93.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

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

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

      3. log-lowering-log.f64 74.0

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

   5. Simplified 74.0%

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

   if -2.00000000000000007e121 < (-.f64 x #s(literal 1 binary64)) < -1.000000000010413

   1. Initial program 84.3%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. +-lowering-+.f64 N/A

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

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

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

      15. sub-neg N/A

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

      16. *-commutative N/A

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

      17. metadata-eval N/A

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

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

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

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

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

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

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

      21. sub-neg N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. distribute-lft-out N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. distribute-neg-in N/A

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

      14. mul-1-neg N/A

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

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

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. +-commutative N/A

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

      19. +-lowering-+.f64 N/A

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

      20. mul-1-neg N/A

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

      21. distribute-neg-in N/A

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

      22. metadata-eval N/A

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

      23. distribute-neg-frac N/A

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

      24. metadata-eval N/A

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

      25. +-lowering-+.f64 N/A

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

      26. /-lowering-/.f64 38.4

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

   8. Simplified 38.4%

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

   9. Taylor expanded in y around 0

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

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

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

       3. associate--r+ N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. distribute-lft-in N/A

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

       7. *-commutative N/A

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

       8. associate-+l+ N/A

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

       9. +-commutative N/A

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

       10. +-commutative N/A

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

       11. associate-+r+ N/A

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

       12. +-commutative N/A

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

       13. associate-+l+ N/A

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

       14. associate--r+ N/A

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

       15. neg-sub0 N/A

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

       16. mul-1-neg N/A

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

       17. remove-double-neg N/A

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

       18. --lowering--.f64 N/A

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

       19. accelerator-lowering-fma.f64 62.5

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

   11. Simplified 62.5%

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

   if -1.000000000010413 < (-.f64 x #s(literal 1 binary64)) < -0.98999999999999999

   1. Initial program 86.1%

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

   3. Taylor expanded in y around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 85.3

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

   5. Simplified 85.3%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-out N/A

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

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

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

      5. +-lowering-+.f64 N/A

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

      6. log-lowering-log.f64 84.6

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

   8. Simplified 84.6%

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

   if -0.98999999999999999 < (-.f64 x #s(literal 1 binary64)) < 1.00000000000000006e63

   1. Initial program 60.6%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. +-lowering-+.f64 N/A

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

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

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

      15. sub-neg N/A

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

      16. *-commutative N/A

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

      17. metadata-eval N/A

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

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

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

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

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

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

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

      21. sub-neg N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 64.3

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

   8. Simplified 64.3%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -2 \cdot 10^{+121}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;-1 + x \leq -1.000000000010413:\\ \;\;\;\;y - \mathsf{fma}\left(y, z, t\right)\\ \mathbf{elif}\;-1 + x \leq -0.99:\\ \;\;\;\;0 - \left(\log y + t\right)\\ \mathbf{elif}\;-1 + x \leq 10^{+63}:\\ \;\;\;\;z \cdot \left(y \cdot \mathsf{fma}\left(y, -0.5, -1\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.5%

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

3. Taylor expanded in y around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. accelerator-lowering-fma.f64 99.8

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 5: 99.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.5%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-out N/A

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

   4. +-commutative N/A

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

   5. metadata-eval N/A

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

   6. sub-neg N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

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

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. +-commutative N/A

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

   13. +-lowering-+.f64 N/A

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

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

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

   15. sub-neg N/A

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

   16. *-commutative N/A

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

   17. metadata-eval N/A

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

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

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

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

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

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

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

   21. sub-neg N/A

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

5. Simplified 99.7%

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

Alternative 6: 95.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + x \leq -1.00000002:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{elif}\;-1 + x \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\left(y - \mathsf{fma}\left(y, z, t\right)\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ -1.0 x) -1.00000002)
   (- (* (log y) (+ -1.0 x)) t)
   (if (<= (+ -1.0 x) 2e+15)
     (- (- y (fma y z t)) (log y))
     (- (* x (log y)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((-1.0 + x) <= -1.00000002) {
		tmp = (log(y) * (-1.0 + x)) - t;
	} else if ((-1.0 + x) <= 2e+15) {
		tmp = (y - fma(y, z, t)) - log(y);
	} else {
		tmp = (x * log(y)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(-1.0 + x) <= -1.00000002)
		tmp = Float64(Float64(log(y) * Float64(-1.0 + x)) - t);
	elseif (Float64(-1.0 + x) <= 2e+15)
		tmp = Float64(Float64(y - fma(y, z, t)) - log(y));
	else
		tmp = Float64(Float64(x * log(y)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(-1.0 + x), $MachinePrecision], -1.00000002], N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(-1.0 + x), $MachinePrecision], 2e+15], N[(N[(y - N[(y * z + t), $MachinePrecision]), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -1.0000000200000001

   1. Initial program 91.3%

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

   3. Taylor expanded in y around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 91.3

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

   5. Simplified 91.3%

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

   if -1.0000000200000001 < (-.f64 x #s(literal 1 binary64)) < 2e15

   1. Initial program 82.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate--l- N/A

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

      5. --lowering--.f64 N/A

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

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

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

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

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

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

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. +-lowering-+.f64 99.5

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. --lowering--.f64 N/A

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

      6. neg-sub0 N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

      14. associate--r+ N/A

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

      15. neg-sub0 N/A

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

      16. mul-1-neg N/A

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

      17. remove-double-neg N/A

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

      18. --lowering--.f64 N/A

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

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

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

      20. log-lowering-log.f64 98.1

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

   8. Simplified 98.1%

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

   if 2e15 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 88.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

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

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

      3. log-lowering-log.f64 87.4

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

   5. Simplified 87.4%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -1.00000002:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{elif}\;-1 + x \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\left(y - \mathsf{fma}\left(y, z, t\right)\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y - \mathsf{fma}\left(y, z, t\right)\\ \mathbf{if}\;-1 + z \leq -1 \cdot 10^{+204}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;-1 + z \leq 10^{+279}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- y (fma y z t))))
   (if (<= (+ -1.0 z) -1e+204)
     t_1
     (if (<= (+ -1.0 z) 1e+279) (- (* (log y) (+ -1.0 x)) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y - fma(y, z, t);
	double tmp;
	if ((-1.0 + z) <= -1e+204) {
		tmp = t_1;
	} else if ((-1.0 + z) <= 1e+279) {
		tmp = (log(y) * (-1.0 + x)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y - fma(y, z, t))
	tmp = 0.0
	if (Float64(-1.0 + z) <= -1e+204)
		tmp = t_1;
	elseif (Float64(-1.0 + z) <= 1e+279)
		tmp = Float64(Float64(log(y) * Float64(-1.0 + x)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y - N[(y * z + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-1.0 + z), $MachinePrecision], -1e+204], t$95$1, If[LessEqual[N[(-1.0 + z), $MachinePrecision], 1e+279], N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z #s(literal 1 binary64)) < -9.99999999999999989e203 or 1.00000000000000006e279 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 36.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. +-lowering-+.f64 N/A

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

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

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

      15. sub-neg N/A

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

      16. *-commutative N/A

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

      17. metadata-eval N/A

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

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

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

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

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

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

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

      21. sub-neg N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. distribute-lft-out N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. distribute-neg-in N/A

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

      14. mul-1-neg N/A

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

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

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. +-commutative N/A

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

      19. +-lowering-+.f64 N/A

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

      20. mul-1-neg N/A

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

      21. distribute-neg-in N/A

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

      22. metadata-eval N/A

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

      23. distribute-neg-frac N/A

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

      24. metadata-eval N/A

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

      25. +-lowering-+.f64 N/A

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

      26. /-lowering-/.f64 11.6

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

   8. Simplified 11.6%

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

   9. Taylor expanded in y around 0

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

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

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

       3. associate--r+ N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. distribute-lft-in N/A

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

       7. *-commutative N/A

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

       8. associate-+l+ N/A

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

       9. +-commutative N/A

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

       10. +-commutative N/A

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

       11. associate-+r+ N/A

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

       12. +-commutative N/A

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

       13. associate-+l+ N/A

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

       14. associate--r+ N/A

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

       15. neg-sub0 N/A

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

       16. mul-1-neg N/A

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

       17. remove-double-neg N/A

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

       18. --lowering--.f64 N/A

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

       19. accelerator-lowering-fma.f64 88.2

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

   11. Simplified 88.2%

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

   if -9.99999999999999989e203 < (-.f64 z #s(literal 1 binary64)) < 1.00000000000000006e279

   1. Initial program 93.9%

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

   3. Taylor expanded in y around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 93.2

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

   5. Simplified 93.2%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + z \leq -1 \cdot 10^{+204}:\\ \;\;\;\;y - \mathsf{fma}\left(y, z, t\right)\\ \mathbf{elif}\;-1 + z \leq 10^{+279}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;y - \mathsf{fma}\left(y, z, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{+46}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;t \leq 125000000:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, 0 - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2.9e+46)
   (- (* x (log y)) t)
   (if (<= t 125000000.0)
     (- (* (log y) (+ -1.0 x)) (* y z))
     (fma (log y) x (- 0.0 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.9e+46) {
		tmp = (x * log(y)) - t;
	} else if (t <= 125000000.0) {
		tmp = (log(y) * (-1.0 + x)) - (y * z);
	} else {
		tmp = fma(log(y), x, (0.0 - t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2.9e+46)
		tmp = Float64(Float64(x * log(y)) - t);
	elseif (t <= 125000000.0)
		tmp = Float64(Float64(log(y) * Float64(-1.0 + x)) - Float64(y * z));
	else
		tmp = fma(log(y), x, Float64(0.0 - t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2.9e+46], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t, 125000000.0], N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x + N[(0.0 - t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.9000000000000002e46

   1. Initial program 96.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

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

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

      3. log-lowering-log.f64 96.7

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

   5. Simplified 96.7%

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

   if -2.9000000000000002e46 < t < 1.25e8

   1. Initial program 78.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate--l- N/A

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

      5. --lowering--.f64 N/A

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

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

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

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

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

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

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. +-lowering-+.f64 99.4

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

   5. Simplified 99.4%

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

   6. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 97.1

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

   8. Simplified 97.1%

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

   if 1.25e8 < t

   1. Initial program 96.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

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

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

      3. log-lowering-log.f64 95.5

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

   5. Simplified 95.5%

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

      1. sub-neg N/A

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

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

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

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

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

      4. neg-sub0 N/A

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

      5. --lowering--.f64 95.5

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

   7. Applied egg-rr 95.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, 0 - t\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{+46}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;t \leq 125000000:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, 0 - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + z \leq -1 \cdot 10^{+146}:\\ \;\;\;\;y - \mathsf{fma}\left(y, z, t\right)\\ \mathbf{elif}\;-1 + z \leq 4 \cdot 10^{+143}:\\ \;\;\;\;0 - \left(\log y + t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), -0.5\right), -1\right) \cdot \left(y \cdot z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ -1.0 z) -1e+146)
   (- y (fma y z t))
   (if (<= (+ -1.0 z) 4e+143)
     (- 0.0 (+ (log y) t))
     (-
      (* (fma y (fma y (fma y -0.25 -0.3333333333333333) -0.5) -1.0) (* y z))
      t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((-1.0 + z) <= -1e+146) {
		tmp = y - fma(y, z, t);
	} else if ((-1.0 + z) <= 4e+143) {
		tmp = 0.0 - (log(y) + t);
	} else {
		tmp = (fma(y, fma(y, fma(y, -0.25, -0.3333333333333333), -0.5), -1.0) * (y * z)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(-1.0 + z) <= -1e+146)
		tmp = Float64(y - fma(y, z, t));
	elseif (Float64(-1.0 + z) <= 4e+143)
		tmp = Float64(0.0 - Float64(log(y) + t));
	else
		tmp = Float64(Float64(fma(y, fma(y, fma(y, -0.25, -0.3333333333333333), -0.5), -1.0) * Float64(y * z)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(-1.0 + z), $MachinePrecision], -1e+146], N[(y - N[(y * z + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-1.0 + z), $MachinePrecision], 4e+143], N[(0.0 - N[(N[Log[y], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * N[(y * N[(y * -0.25 + -0.3333333333333333), $MachinePrecision] + -0.5), $MachinePrecision] + -1.0), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 z #s(literal 1 binary64)) < -9.99999999999999934e145

   1. Initial program 43.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. +-lowering-+.f64 N/A

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

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

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

      15. sub-neg N/A

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

      16. *-commutative N/A

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

      17. metadata-eval N/A

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

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

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

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

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

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

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

      21. sub-neg N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. distribute-lft-out N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. distribute-neg-in N/A

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

      14. mul-1-neg N/A

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

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

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. +-commutative N/A

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

      19. +-lowering-+.f64 N/A

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

      20. mul-1-neg N/A

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

      21. distribute-neg-in N/A

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

      22. metadata-eval N/A

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

      23. distribute-neg-frac N/A

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

      24. metadata-eval N/A

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

      25. +-lowering-+.f64 N/A

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

      26. /-lowering-/.f64 16.3

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

   8. Simplified 16.3%

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

   9. Taylor expanded in y around 0

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

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

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

       3. associate--r+ N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. distribute-lft-in N/A

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

       7. *-commutative N/A

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

       8. associate-+l+ N/A

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

       9. +-commutative N/A

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

       10. +-commutative N/A

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

       11. associate-+r+ N/A

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

       12. +-commutative N/A

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

       13. associate-+l+ N/A

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

       14. associate--r+ N/A

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

       15. neg-sub0 N/A

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

       16. mul-1-neg N/A

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

       17. remove-double-neg N/A

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

       18. --lowering--.f64 N/A

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

       19. accelerator-lowering-fma.f64 74.9

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

   11. Simplified 74.9%

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

   if -9.99999999999999934e145 < (-.f64 z #s(literal 1 binary64)) < 4.0000000000000001e143

   1. Initial program 97.8%

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

   3. Taylor expanded in y around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 97.8

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

   5. Simplified 97.8%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-out N/A

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

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

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

      5. +-lowering-+.f64 N/A

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

      6. log-lowering-log.f64 63.0

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

   8. Simplified 63.0%

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

   if 4.0000000000000001e143 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 70.7%

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

   3. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

   5. Simplified 99.8%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

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

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-lowering-*.f64 72.8

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

   8. Simplified 72.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), -0.5\right), -1\right) \cdot \left(y \cdot z\right)} - t \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + z \leq -1 \cdot 10^{+146}:\\ \;\;\;\;y - \mathsf{fma}\left(y, z, t\right)\\ \mathbf{elif}\;-1 + z \leq 4 \cdot 10^{+143}:\\ \;\;\;\;0 - \left(\log y + t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), -0.5\right), -1\right) \cdot \left(y \cdot z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{+15}:\\ \;\;\;\;y - \mathsf{fma}\left(y, z, t\right)\\ \mathbf{elif}\;t \leq 1250:\\ \;\;\;\;\log y \cdot \left(-1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(\frac{\mathsf{fma}\left(y, 0.5, 1\right)}{z} - \mathsf{fma}\left(y, 0.5, 1\right)\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.15e+15)
   (- y (fma y z t))
   (if (<= t 1250.0)
     (* (log y) (+ -1.0 x))
     (- (* z (* y (- (/ (fma y 0.5 1.0) z) (fma y 0.5 1.0)))) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.15e+15) {
		tmp = y - fma(y, z, t);
	} else if (t <= 1250.0) {
		tmp = log(y) * (-1.0 + x);
	} else {
		tmp = (z * (y * ((fma(y, 0.5, 1.0) / z) - fma(y, 0.5, 1.0)))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.15e+15)
		tmp = Float64(y - fma(y, z, t));
	elseif (t <= 1250.0)
		tmp = Float64(log(y) * Float64(-1.0 + x));
	else
		tmp = Float64(Float64(z * Float64(y * Float64(Float64(fma(y, 0.5, 1.0) / z) - fma(y, 0.5, 1.0)))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.15e+15], N[(y - N[(y * z + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1250.0], N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(y * N[(N[(N[(y * 0.5 + 1.0), $MachinePrecision] / z), $MachinePrecision] - N[(y * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.15e15

   1. Initial program 89.5%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. +-lowering-+.f64 N/A

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

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

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

      15. sub-neg N/A

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

      16. *-commutative N/A

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

      17. metadata-eval N/A

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

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

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

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

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

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

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

      21. sub-neg N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. distribute-lft-out N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. distribute-neg-in N/A

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

      14. mul-1-neg N/A

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

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

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. +-commutative N/A

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

      19. +-lowering-+.f64 N/A

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

      20. mul-1-neg N/A

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

      21. distribute-neg-in N/A

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

      22. metadata-eval N/A

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

      23. distribute-neg-frac N/A

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

      24. metadata-eval N/A

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

      25. +-lowering-+.f64 N/A

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

      26. /-lowering-/.f64 63.4

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

   8. Simplified 63.4%

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

   9. Taylor expanded in y around 0

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

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

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

       3. associate--r+ N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. distribute-lft-in N/A

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

       7. *-commutative N/A

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

       8. associate-+l+ N/A

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

       9. +-commutative N/A

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

       10. +-commutative N/A

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

       11. associate-+r+ N/A

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

       12. +-commutative N/A

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

       13. associate-+l+ N/A

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

       14. associate--r+ N/A

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

       15. neg-sub0 N/A

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

       16. mul-1-neg N/A

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

       17. remove-double-neg N/A

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

       18. --lowering--.f64 N/A

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

       19. accelerator-lowering-fma.f64 83.4

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

   11. Simplified 83.4%

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

   if -1.15e15 < t < 1250

   1. Initial program 81.7%

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

   3. Taylor expanded in y around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 81.0

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

   5. Simplified 81.0%

      \[\leadsto \color{blue}{\log y \cdot \left(-1 + x\right)} - t \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\log y \cdot \left(x - 1\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\log y \cdot \left(x - 1\right)} \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \color{blue}{\log y} \cdot \left(x - 1\right) \]

      3. sub-neg N/A

         \[\leadsto \log y \cdot \color{blue}{\left(x + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      4. metadata-eval N/A

         \[\leadsto \log y \cdot \left(x + \color{blue}{-1}\right) \]

      5. +-lowering-+.f64 80.1

         \[\leadsto \log y \cdot \color{blue}{\left(x + -1\right)} \]

   8. Simplified 80.1%

      \[\leadsto \color{blue}{\log y \cdot \left(x + -1\right)} \]

   if 1250 < t

   1. Initial program 95.0%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(y \cdot \left(-1 \cdot \left(z - 1\right) + \frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot \left(z - 1\right) + \frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right)\right) \cdot y} + \log y \cdot \left(x - 1\right)\right) - t \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(-1 \cdot \left(z - 1\right) + \color{blue}{\left(\frac{-1}{2} \cdot y\right) \cdot \left(z - 1\right)}\right) \cdot y + \log y \cdot \left(x - 1\right)\right) - t \]

      3. distribute-rgt-out N/A

         \[\leadsto \left(\color{blue}{\left(\left(z - 1\right) \cdot \left(-1 + \frac{-1}{2} \cdot y\right)\right)} \cdot y + \log y \cdot \left(x - 1\right)\right) - t \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\left(z - 1\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot y + -1\right)}\right) \cdot y + \log y \cdot \left(x - 1\right)\right) - t \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(\left(z - 1\right) \cdot \left(\frac{-1}{2} \cdot y + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \cdot y + \log y \cdot \left(x - 1\right)\right) - t \]

      6. sub-neg N/A

         \[\leadsto \left(\left(\left(z - 1\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot y - 1\right)}\right) \cdot y + \log y \cdot \left(x - 1\right)\right) - t \]

      7. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\left(z - 1\right) \cdot \left(\left(\frac{-1}{2} \cdot y - 1\right) \cdot y\right)} + \log y \cdot \left(x - 1\right)\right) - t \]

      8. *-commutative N/A

         \[\leadsto \left(\left(z - 1\right) \cdot \color{blue}{\left(y \cdot \left(\frac{-1}{2} \cdot y - 1\right)\right)} + \log y \cdot \left(x - 1\right)\right) - t \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z - 1, y \cdot \left(\frac{-1}{2} \cdot y - 1\right), \log y \cdot \left(x - 1\right)\right)} - t \]

      10. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z + \left(\mathsf{neg}\left(1\right)\right)}, y \cdot \left(\frac{-1}{2} \cdot y - 1\right), \log y \cdot \left(x - 1\right)\right) - t \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z + \color{blue}{-1}, y \cdot \left(\frac{-1}{2} \cdot y - 1\right), \log y \cdot \left(x - 1\right)\right) - t \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-1 + z}, y \cdot \left(\frac{-1}{2} \cdot y - 1\right), \log y \cdot \left(x - 1\right)\right) - t \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-1 + z}, y \cdot \left(\frac{-1}{2} \cdot y - 1\right), \log y \cdot \left(x - 1\right)\right) - t \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-1 + z, \color{blue}{y \cdot \left(\frac{-1}{2} \cdot y - 1\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

      15. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \color{blue}{\left(\frac{-1}{2} \cdot y + \left(\mathsf{neg}\left(1\right)\right)\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \left(\color{blue}{y \cdot \frac{-1}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right), \log y \cdot \left(x - 1\right)\right) - t \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \left(y \cdot \frac{-1}{2} + \color{blue}{-1}\right), \log y \cdot \left(x - 1\right)\right) - t \]

      18. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \color{blue}{\mathsf{fma}\left(y, \frac{-1}{2}, -1\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \mathsf{fma}\left(y, \frac{-1}{2}, -1\right), \color{blue}{\log y \cdot \left(x - 1\right)}\right) - t \]

      20. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \mathsf{fma}\left(y, \frac{-1}{2}, -1\right), \color{blue}{\log y} \cdot \left(x - 1\right)\right) - t \]

      21. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \mathsf{fma}\left(y, \frac{-1}{2}, -1\right), \log y \cdot \color{blue}{\left(x + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) - t \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1 + z, y \cdot \mathsf{fma}\left(y, -0.5, -1\right), \log y \cdot \left(-1 + x\right)\right)} - t \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{{y}^{2} \cdot \left(-1 \cdot \frac{z - 1}{y} + \frac{-1}{2} \cdot \left(z - 1\right)\right)} - t \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot \left(-1 \cdot \frac{z - 1}{y} + \frac{-1}{2} \cdot \left(z - 1\right)\right)} - t \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot \left(-1 \cdot \frac{z - 1}{y} + \frac{-1}{2} \cdot \left(z - 1\right)\right) - t \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot \left(-1 \cdot \frac{z - 1}{y} + \frac{-1}{2} \cdot \left(z - 1\right)\right) - t \]

      4. +-commutative N/A

         \[\leadsto \left(y \cdot y\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \left(z - 1\right) + -1 \cdot \frac{z - 1}{y}\right)} - t \]

      5. *-commutative N/A

         \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{\left(z - 1\right) \cdot \frac{-1}{2}} + -1 \cdot \frac{z - 1}{y}\right) - t \]

      6. associate-*r/ N/A

         \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \frac{-1}{2} + \color{blue}{\frac{-1 \cdot \left(z - 1\right)}{y}}\right) - t \]

      7. *-commutative N/A

         \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \frac{-1}{2} + \frac{\color{blue}{\left(z - 1\right) \cdot -1}}{y}\right) - t \]

      8. associate-/l* N/A

         \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \frac{-1}{2} + \color{blue}{\left(z - 1\right) \cdot \frac{-1}{y}}\right) - t \]

      9. distribute-lft-out N/A

         \[\leadsto \left(y \cdot y\right) \cdot \color{blue}{\left(\left(z - 1\right) \cdot \left(\frac{-1}{2} + \frac{-1}{y}\right)\right)} - t \]

      10. metadata-eval N/A

          \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \frac{-1}{y}\right)\right) - t \]

      11. metadata-eval N/A

          \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{y}\right)\right) - t \]

      12. distribute-neg-frac N/A

          \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{y}\right)\right)}\right)\right) - t \]

      13. distribute-neg-in N/A

          \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} + \frac{1}{y}\right)\right)\right)}\right) - t \]

      14. mul-1-neg N/A

          \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \color{blue}{\left(-1 \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)}\right) - t \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \left(y \cdot y\right) \cdot \color{blue}{\left(\left(z - 1\right) \cdot \left(-1 \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)\right)} - t \]

      16. sub-neg N/A

          \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{\left(z + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(-1 \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)\right) - t \]

      17. metadata-eval N/A

          \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z + \color{blue}{-1}\right) \cdot \left(-1 \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)\right) - t \]

      18. +-commutative N/A

          \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{\left(-1 + z\right)} \cdot \left(-1 \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)\right) - t \]

      19. +-lowering-+.f64 N/A

          \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{\left(-1 + z\right)} \cdot \left(-1 \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)\right) - t \]

      20. mul-1-neg N/A

          \[\leadsto \left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} + \frac{1}{y}\right)\right)\right)}\right) - t \]

      21. distribute-neg-in N/A

          \[\leadsto \left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{y}\right)\right)\right)}\right) - t \]

      22. metadata-eval N/A

          \[\leadsto \left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \left(\color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(\frac{1}{y}\right)\right)\right)\right) - t \]

      23. distribute-neg-frac N/A

          \[\leadsto \left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \left(\frac{-1}{2} + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{y}}\right)\right) - t \]

      24. metadata-eval N/A

          \[\leadsto \left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \left(\frac{-1}{2} + \frac{\color{blue}{-1}}{y}\right)\right) - t \]

      25. +-lowering-+.f64 N/A

          \[\leadsto \left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \color{blue}{\left(\frac{-1}{2} + \frac{-1}{y}\right)}\right) - t \]

      26. /-lowering-/.f64 49.2

          \[\leadsto \left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \left(-0.5 + \color{blue}{\frac{-1}{y}}\right)\right) - t \]

   8. Simplified 49.2%

      \[\leadsto \color{blue}{\left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \left(-0.5 + \frac{-1}{y}\right)\right)} - t \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(-1 \cdot \left({y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right) + \frac{{y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{y}\right)}{z}\right)} - t \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{z \cdot \left(-1 \cdot \left({y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right) + \frac{{y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{y}\right)}{z}\right)} - t \]

       2. +-commutative N/A

          \[\leadsto z \cdot \color{blue}{\left(\frac{{y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{y}\right)}{z} + -1 \cdot \left({y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)\right)} - t \]

       3. mul-1-neg N/A

          \[\leadsto z \cdot \left(\frac{{y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{y}\right)}{z} + \color{blue}{\left(\mathsf{neg}\left({y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)\right)}\right) - t \]

       4. unsub-neg N/A

          \[\leadsto z \cdot \color{blue}{\left(\frac{{y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{y}\right)}{z} - {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)} - t \]

       5. associate-/l* N/A

          \[\leadsto z \cdot \left(\color{blue}{{y}^{2} \cdot \frac{\frac{1}{2} + \frac{1}{y}}{z}} - {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right) - t \]

       6. unpow2 N/A

          \[\leadsto z \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{\frac{1}{2} + \frac{1}{y}}{z} - {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right) - t \]

       7. associate-*l* N/A

          \[\leadsto z \cdot \left(\color{blue}{y \cdot \left(y \cdot \frac{\frac{1}{2} + \frac{1}{y}}{z}\right)} - {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right) - t \]

       8. unpow2 N/A

          \[\leadsto z \cdot \left(y \cdot \left(y \cdot \frac{\frac{1}{2} + \frac{1}{y}}{z}\right) - \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right) - t \]

       9. associate-*l* N/A

          \[\leadsto z \cdot \left(y \cdot \left(y \cdot \frac{\frac{1}{2} + \frac{1}{y}}{z}\right) - \color{blue}{y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)}\right) - t \]

       10. distribute-lft-out-- N/A

           \[\leadsto z \cdot \color{blue}{\left(y \cdot \left(y \cdot \frac{\frac{1}{2} + \frac{1}{y}}{z} - y \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)\right)} - t \]

       11. *-lowering-*.f64 N/A

           \[\leadsto z \cdot \color{blue}{\left(y \cdot \left(y \cdot \frac{\frac{1}{2} + \frac{1}{y}}{z} - y \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)\right)} - t \]

       12. --lowering--.f64 N/A

           \[\leadsto z \cdot \left(y \cdot \color{blue}{\left(y \cdot \frac{\frac{1}{2} + \frac{1}{y}}{z} - y \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)}\right) - t \]

   11. Simplified 78.9%

       \[\leadsto \color{blue}{z \cdot \left(y \cdot \left(\frac{\mathsf{fma}\left(y, 0.5, 1\right)}{z} - \mathsf{fma}\left(y, 0.5, 1\right)\right)\right)} - t \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{+15}:\\ \;\;\;\;y - \mathsf{fma}\left(y, z, t\right)\\ \mathbf{elif}\;t \leq 1250:\\ \;\;\;\;\log y \cdot \left(-1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(\frac{\mathsf{fma}\left(y, 0.5, 1\right)}{z} - \mathsf{fma}\left(y, 0.5, 1\right)\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \log y \cdot \left(-1 + x\right) - \mathsf{fma}\left(y, -1 + z, t\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (* (log y) (+ -1.0 x)) (fma y (+ -1.0 z) t)))

C

double code(double x, double y, double z, double t) {
	return (log(y) * (-1.0 + x)) - fma(y, (-1.0 + z), t);
}

Julia

function code(x, y, z, t)
	return Float64(Float64(log(y) * Float64(-1.0 + x)) - fma(y, Float64(-1.0 + z), t))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(y * N[(-1.0 + z), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.5%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + \log y \cdot \left(x - 1\right)\right) - t} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(\log y \cdot \left(x - 1\right) + -1 \cdot \left(y \cdot \left(z - 1\right)\right)\right)} - t \]

   2. mul-1-neg N/A

      \[\leadsto \left(\log y \cdot \left(x - 1\right) + \color{blue}{\left(\mathsf{neg}\left(y \cdot \left(z - 1\right)\right)\right)}\right) - t \]

   3. unsub-neg N/A

      \[\leadsto \color{blue}{\left(\log y \cdot \left(x - 1\right) - y \cdot \left(z - 1\right)\right)} - t \]

   4. associate--l- N/A

      \[\leadsto \color{blue}{\log y \cdot \left(x - 1\right) - \left(y \cdot \left(z - 1\right) + t\right)} \]

   5. --lowering--.f64 N/A

      \[\leadsto \color{blue}{\log y \cdot \left(x - 1\right) - \left(y \cdot \left(z - 1\right) + t\right)} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\log y \cdot \left(x - 1\right)} - \left(y \cdot \left(z - 1\right) + t\right) \]

   7. log-lowering-log.f64 N/A

      \[\leadsto \color{blue}{\log y} \cdot \left(x - 1\right) - \left(y \cdot \left(z - 1\right) + t\right) \]

   8. sub-neg N/A

      \[\leadsto \log y \cdot \color{blue}{\left(x + \left(\mathsf{neg}\left(1\right)\right)\right)} - \left(y \cdot \left(z - 1\right) + t\right) \]

   9. metadata-eval N/A

      \[\leadsto \log y \cdot \left(x + \color{blue}{-1}\right) - \left(y \cdot \left(z - 1\right) + t\right) \]

   10. +-commutative N/A

       \[\leadsto \log y \cdot \color{blue}{\left(-1 + x\right)} - \left(y \cdot \left(z - 1\right) + t\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \log y \cdot \color{blue}{\left(-1 + x\right)} - \left(y \cdot \left(z - 1\right) + t\right) \]

   12. accelerator-lowering-fma.f64 N/A

       \[\leadsto \log y \cdot \left(-1 + x\right) - \color{blue}{\mathsf{fma}\left(y, z - 1, t\right)} \]

   13. sub-neg N/A

       \[\leadsto \log y \cdot \left(-1 + x\right) - \mathsf{fma}\left(y, \color{blue}{z + \left(\mathsf{neg}\left(1\right)\right)}, t\right) \]

   14. metadata-eval N/A

       \[\leadsto \log y \cdot \left(-1 + x\right) - \mathsf{fma}\left(y, z + \color{blue}{-1}, t\right) \]

   15. +-commutative N/A

       \[\leadsto \log y \cdot \left(-1 + x\right) - \mathsf{fma}\left(y, \color{blue}{-1 + z}, t\right) \]

   16. +-lowering-+.f64 99.5

       \[\leadsto \log y \cdot \left(-1 + x\right) - \mathsf{fma}\left(y, \color{blue}{-1 + z}, t\right) \]

5. Simplified 99.5%

   \[\leadsto \color{blue}{\log y \cdot \left(-1 + x\right) - \mathsf{fma}\left(y, -1 + z, t\right)} \]
6. Add Preprocessing

Alternative 12: 46.7% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ z \cdot \left(y \cdot \left(\frac{\mathsf{fma}\left(y, 0.5, 1\right)}{z} - \mathsf{fma}\left(y, 0.5, 1\right)\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (* z (* y (- (/ (fma y 0.5 1.0) z) (fma y 0.5 1.0)))) t))

C

double code(double x, double y, double z, double t) {
	return (z * (y * ((fma(y, 0.5, 1.0) / z) - fma(y, 0.5, 1.0)))) - t;
}

Julia

function code(x, y, z, t)
	return Float64(Float64(z * Float64(y * Float64(Float64(fma(y, 0.5, 1.0) / z) - fma(y, 0.5, 1.0)))) - t)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(z * N[(y * N[(N[(N[(y * 0.5 + 1.0), $MachinePrecision] / z), $MachinePrecision] - N[(y * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 86.5%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(y \cdot \left(-1 \cdot \left(z - 1\right) + \frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(-1 \cdot \left(z - 1\right) + \frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right)\right) \cdot y} + \log y \cdot \left(x - 1\right)\right) - t \]

   2. associate-*r* N/A

      \[\leadsto \left(\left(-1 \cdot \left(z - 1\right) + \color{blue}{\left(\frac{-1}{2} \cdot y\right) \cdot \left(z - 1\right)}\right) \cdot y + \log y \cdot \left(x - 1\right)\right) - t \]

   3. distribute-rgt-out N/A

      \[\leadsto \left(\color{blue}{\left(\left(z - 1\right) \cdot \left(-1 + \frac{-1}{2} \cdot y\right)\right)} \cdot y + \log y \cdot \left(x - 1\right)\right) - t \]

   4. +-commutative N/A

      \[\leadsto \left(\left(\left(z - 1\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot y + -1\right)}\right) \cdot y + \log y \cdot \left(x - 1\right)\right) - t \]

   5. metadata-eval N/A

      \[\leadsto \left(\left(\left(z - 1\right) \cdot \left(\frac{-1}{2} \cdot y + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \cdot y + \log y \cdot \left(x - 1\right)\right) - t \]

   6. sub-neg N/A

      \[\leadsto \left(\left(\left(z - 1\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot y - 1\right)}\right) \cdot y + \log y \cdot \left(x - 1\right)\right) - t \]

   7. associate-*l* N/A

      \[\leadsto \left(\color{blue}{\left(z - 1\right) \cdot \left(\left(\frac{-1}{2} \cdot y - 1\right) \cdot y\right)} + \log y \cdot \left(x - 1\right)\right) - t \]

   8. *-commutative N/A

      \[\leadsto \left(\left(z - 1\right) \cdot \color{blue}{\left(y \cdot \left(\frac{-1}{2} \cdot y - 1\right)\right)} + \log y \cdot \left(x - 1\right)\right) - t \]

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(z - 1, y \cdot \left(\frac{-1}{2} \cdot y - 1\right), \log y \cdot \left(x - 1\right)\right)} - t \]

   10. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{z + \left(\mathsf{neg}\left(1\right)\right)}, y \cdot \left(\frac{-1}{2} \cdot y - 1\right), \log y \cdot \left(x - 1\right)\right) - t \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(z + \color{blue}{-1}, y \cdot \left(\frac{-1}{2} \cdot y - 1\right), \log y \cdot \left(x - 1\right)\right) - t \]

   12. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{-1 + z}, y \cdot \left(\frac{-1}{2} \cdot y - 1\right), \log y \cdot \left(x - 1\right)\right) - t \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{-1 + z}, y \cdot \left(\frac{-1}{2} \cdot y - 1\right), \log y \cdot \left(x - 1\right)\right) - t \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, \color{blue}{y \cdot \left(\frac{-1}{2} \cdot y - 1\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

   15. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \color{blue}{\left(\frac{-1}{2} \cdot y + \left(\mathsf{neg}\left(1\right)\right)\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

   16. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \left(\color{blue}{y \cdot \frac{-1}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right), \log y \cdot \left(x - 1\right)\right) - t \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \left(y \cdot \frac{-1}{2} + \color{blue}{-1}\right), \log y \cdot \left(x - 1\right)\right) - t \]

   18. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \color{blue}{\mathsf{fma}\left(y, \frac{-1}{2}, -1\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

   19. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \mathsf{fma}\left(y, \frac{-1}{2}, -1\right), \color{blue}{\log y \cdot \left(x - 1\right)}\right) - t \]

   20. log-lowering-log.f64 N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \mathsf{fma}\left(y, \frac{-1}{2}, -1\right), \color{blue}{\log y} \cdot \left(x - 1\right)\right) - t \]

   21. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \mathsf{fma}\left(y, \frac{-1}{2}, -1\right), \log y \cdot \color{blue}{\left(x + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) - t \]

5. Simplified 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-1 + z, y \cdot \mathsf{fma}\left(y, -0.5, -1\right), \log y \cdot \left(-1 + x\right)\right)} - t \]

6. Taylor expanded in y around inf

   \[\leadsto \color{blue}{{y}^{2} \cdot \left(-1 \cdot \frac{z - 1}{y} + \frac{-1}{2} \cdot \left(z - 1\right)\right)} - t \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{{y}^{2} \cdot \left(-1 \cdot \frac{z - 1}{y} + \frac{-1}{2} \cdot \left(z - 1\right)\right)} - t \]

   2. unpow2 N/A

      \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot \left(-1 \cdot \frac{z - 1}{y} + \frac{-1}{2} \cdot \left(z - 1\right)\right) - t \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot \left(-1 \cdot \frac{z - 1}{y} + \frac{-1}{2} \cdot \left(z - 1\right)\right) - t \]

   4. +-commutative N/A

      \[\leadsto \left(y \cdot y\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \left(z - 1\right) + -1 \cdot \frac{z - 1}{y}\right)} - t \]

   5. *-commutative N/A

      \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{\left(z - 1\right) \cdot \frac{-1}{2}} + -1 \cdot \frac{z - 1}{y}\right) - t \]

   6. associate-*r/ N/A

      \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \frac{-1}{2} + \color{blue}{\frac{-1 \cdot \left(z - 1\right)}{y}}\right) - t \]

   7. *-commutative N/A

      \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \frac{-1}{2} + \frac{\color{blue}{\left(z - 1\right) \cdot -1}}{y}\right) - t \]

   8. associate-/l* N/A

      \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \frac{-1}{2} + \color{blue}{\left(z - 1\right) \cdot \frac{-1}{y}}\right) - t \]

   9. distribute-lft-out N/A

      \[\leadsto \left(y \cdot y\right) \cdot \color{blue}{\left(\left(z - 1\right) \cdot \left(\frac{-1}{2} + \frac{-1}{y}\right)\right)} - t \]

   10. metadata-eval N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \frac{-1}{y}\right)\right) - t \]

   11. metadata-eval N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{y}\right)\right) - t \]

   12. distribute-neg-frac N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{y}\right)\right)}\right)\right) - t \]

   13. distribute-neg-in N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} + \frac{1}{y}\right)\right)\right)}\right) - t \]

   14. mul-1-neg N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \color{blue}{\left(-1 \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)}\right) - t \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \left(y \cdot y\right) \cdot \color{blue}{\left(\left(z - 1\right) \cdot \left(-1 \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)\right)} - t \]

   16. sub-neg N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{\left(z + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(-1 \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)\right) - t \]

   17. metadata-eval N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z + \color{blue}{-1}\right) \cdot \left(-1 \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)\right) - t \]

   18. +-commutative N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{\left(-1 + z\right)} \cdot \left(-1 \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)\right) - t \]

   19. +-lowering-+.f64 N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{\left(-1 + z\right)} \cdot \left(-1 \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)\right) - t \]

   20. mul-1-neg N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} + \frac{1}{y}\right)\right)\right)}\right) - t \]

   21. distribute-neg-in N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{y}\right)\right)\right)}\right) - t \]

   22. metadata-eval N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \left(\color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(\frac{1}{y}\right)\right)\right)\right) - t \]

   23. distribute-neg-frac N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \left(\frac{-1}{2} + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{y}}\right)\right) - t \]

   24. metadata-eval N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \left(\frac{-1}{2} + \frac{\color{blue}{-1}}{y}\right)\right) - t \]

   25. +-lowering-+.f64 N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \color{blue}{\left(\frac{-1}{2} + \frac{-1}{y}\right)}\right) - t \]

   26. /-lowering-/.f64 32.4

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \left(-0.5 + \color{blue}{\frac{-1}{y}}\right)\right) - t \]

8. Simplified 32.4%

   \[\leadsto \color{blue}{\left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \left(-0.5 + \frac{-1}{y}\right)\right)} - t \]

9. Taylor expanded in z around inf

   \[\leadsto \color{blue}{z \cdot \left(-1 \cdot \left({y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right) + \frac{{y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{y}\right)}{z}\right)} - t \]
10. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \color{blue}{z \cdot \left(-1 \cdot \left({y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right) + \frac{{y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{y}\right)}{z}\right)} - t \]

    2. +-commutative N/A

       \[\leadsto z \cdot \color{blue}{\left(\frac{{y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{y}\right)}{z} + -1 \cdot \left({y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)\right)} - t \]

    3. mul-1-neg N/A

       \[\leadsto z \cdot \left(\frac{{y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{y}\right)}{z} + \color{blue}{\left(\mathsf{neg}\left({y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)\right)}\right) - t \]

    4. unsub-neg N/A

       \[\leadsto z \cdot \color{blue}{\left(\frac{{y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{y}\right)}{z} - {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)} - t \]

    5. associate-/l* N/A

       \[\leadsto z \cdot \left(\color{blue}{{y}^{2} \cdot \frac{\frac{1}{2} + \frac{1}{y}}{z}} - {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right) - t \]

    6. unpow2 N/A

       \[\leadsto z \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{\frac{1}{2} + \frac{1}{y}}{z} - {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right) - t \]

    7. associate-*l* N/A

       \[\leadsto z \cdot \left(\color{blue}{y \cdot \left(y \cdot \frac{\frac{1}{2} + \frac{1}{y}}{z}\right)} - {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right) - t \]

    8. unpow2 N/A

       \[\leadsto z \cdot \left(y \cdot \left(y \cdot \frac{\frac{1}{2} + \frac{1}{y}}{z}\right) - \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right) - t \]

    9. associate-*l* N/A

       \[\leadsto z \cdot \left(y \cdot \left(y \cdot \frac{\frac{1}{2} + \frac{1}{y}}{z}\right) - \color{blue}{y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)}\right) - t \]

    10. distribute-lft-out-- N/A

        \[\leadsto z \cdot \color{blue}{\left(y \cdot \left(y \cdot \frac{\frac{1}{2} + \frac{1}{y}}{z} - y \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)\right)} - t \]

    11. *-lowering-*.f64 N/A

        \[\leadsto z \cdot \color{blue}{\left(y \cdot \left(y \cdot \frac{\frac{1}{2} + \frac{1}{y}}{z} - y \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)\right)} - t \]

    12. --lowering--.f64 N/A

        \[\leadsto z \cdot \left(y \cdot \color{blue}{\left(y \cdot \frac{\frac{1}{2} + \frac{1}{y}}{z} - y \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)}\right) - t \]

11. Simplified 48.8%

    \[\leadsto \color{blue}{z \cdot \left(y \cdot \left(\frac{\mathsf{fma}\left(y, 0.5, 1\right)}{z} - \mathsf{fma}\left(y, 0.5, 1\right)\right)\right)} - t \]
12. Add Preprocessing

Alternative 13: 46.6% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \left(-0.5 + \frac{-1}{y}\right) \cdot \left(y \cdot \left(y \cdot \left(-1 + z\right)\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (* (+ -0.5 (/ -1.0 y)) (* y (* y (+ -1.0 z)))) t))

C

double code(double x, double y, double z, double t) {
	return ((-0.5 + (-1.0 / y)) * (y * (y * (-1.0 + z)))) - t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((-0.5d0) + ((-1.0d0) / y)) * (y * (y * ((-1.0d0) + z)))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((-0.5 + (-1.0 / y)) * (y * (y * (-1.0 + z)))) - t;
}

Python

def code(x, y, z, t):
	return ((-0.5 + (-1.0 / y)) * (y * (y * (-1.0 + z)))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(-0.5 + Float64(-1.0 / y)) * Float64(y * Float64(y * Float64(-1.0 + z)))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((-0.5 + (-1.0 / y)) * (y * (y * (-1.0 + z)))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(-0.5 + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * N[(-1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 86.5%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(y \cdot \left(-1 \cdot \left(z - 1\right) + \frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(-1 \cdot \left(z - 1\right) + \frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right)\right) \cdot y} + \log y \cdot \left(x - 1\right)\right) - t \]

   2. associate-*r* N/A

      \[\leadsto \left(\left(-1 \cdot \left(z - 1\right) + \color{blue}{\left(\frac{-1}{2} \cdot y\right) \cdot \left(z - 1\right)}\right) \cdot y + \log y \cdot \left(x - 1\right)\right) - t \]

   3. distribute-rgt-out N/A

      \[\leadsto \left(\color{blue}{\left(\left(z - 1\right) \cdot \left(-1 + \frac{-1}{2} \cdot y\right)\right)} \cdot y + \log y \cdot \left(x - 1\right)\right) - t \]

   4. +-commutative N/A

      \[\leadsto \left(\left(\left(z - 1\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot y + -1\right)}\right) \cdot y + \log y \cdot \left(x - 1\right)\right) - t \]

   5. metadata-eval N/A

      \[\leadsto \left(\left(\left(z - 1\right) \cdot \left(\frac{-1}{2} \cdot y + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \cdot y + \log y \cdot \left(x - 1\right)\right) - t \]

   6. sub-neg N/A

      \[\leadsto \left(\left(\left(z - 1\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot y - 1\right)}\right) \cdot y + \log y \cdot \left(x - 1\right)\right) - t \]

   7. associate-*l* N/A

      \[\leadsto \left(\color{blue}{\left(z - 1\right) \cdot \left(\left(\frac{-1}{2} \cdot y - 1\right) \cdot y\right)} + \log y \cdot \left(x - 1\right)\right) - t \]

   8. *-commutative N/A

      \[\leadsto \left(\left(z - 1\right) \cdot \color{blue}{\left(y \cdot \left(\frac{-1}{2} \cdot y - 1\right)\right)} + \log y \cdot \left(x - 1\right)\right) - t \]

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(z - 1, y \cdot \left(\frac{-1}{2} \cdot y - 1\right), \log y \cdot \left(x - 1\right)\right)} - t \]

   10. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{z + \left(\mathsf{neg}\left(1\right)\right)}, y \cdot \left(\frac{-1}{2} \cdot y - 1\right), \log y \cdot \left(x - 1\right)\right) - t \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(z + \color{blue}{-1}, y \cdot \left(\frac{-1}{2} \cdot y - 1\right), \log y \cdot \left(x - 1\right)\right) - t \]

   12. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{-1 + z}, y \cdot \left(\frac{-1}{2} \cdot y - 1\right), \log y \cdot \left(x - 1\right)\right) - t \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{-1 + z}, y \cdot \left(\frac{-1}{2} \cdot y - 1\right), \log y \cdot \left(x - 1\right)\right) - t \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, \color{blue}{y \cdot \left(\frac{-1}{2} \cdot y - 1\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

   15. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \color{blue}{\left(\frac{-1}{2} \cdot y + \left(\mathsf{neg}\left(1\right)\right)\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

   16. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \left(\color{blue}{y \cdot \frac{-1}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right), \log y \cdot \left(x - 1\right)\right) - t \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \left(y \cdot \frac{-1}{2} + \color{blue}{-1}\right), \log y \cdot \left(x - 1\right)\right) - t \]

   18. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \color{blue}{\mathsf{fma}\left(y, \frac{-1}{2}, -1\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

   19. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \mathsf{fma}\left(y, \frac{-1}{2}, -1\right), \color{blue}{\log y \cdot \left(x - 1\right)}\right) - t \]

   20. log-lowering-log.f64 N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \mathsf{fma}\left(y, \frac{-1}{2}, -1\right), \color{blue}{\log y} \cdot \left(x - 1\right)\right) - t \]

   21. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \mathsf{fma}\left(y, \frac{-1}{2}, -1\right), \log y \cdot \color{blue}{\left(x + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) - t \]

5. Simplified 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-1 + z, y \cdot \mathsf{fma}\left(y, -0.5, -1\right), \log y \cdot \left(-1 + x\right)\right)} - t \]

6. Taylor expanded in y around inf

   \[\leadsto \color{blue}{{y}^{2} \cdot \left(-1 \cdot \frac{z - 1}{y} + \frac{-1}{2} \cdot \left(z - 1\right)\right)} - t \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{{y}^{2} \cdot \left(-1 \cdot \frac{z - 1}{y} + \frac{-1}{2} \cdot \left(z - 1\right)\right)} - t \]

   2. unpow2 N/A

      \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot \left(-1 \cdot \frac{z - 1}{y} + \frac{-1}{2} \cdot \left(z - 1\right)\right) - t \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot \left(-1 \cdot \frac{z - 1}{y} + \frac{-1}{2} \cdot \left(z - 1\right)\right) - t \]

   4. +-commutative N/A

      \[\leadsto \left(y \cdot y\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \left(z - 1\right) + -1 \cdot \frac{z - 1}{y}\right)} - t \]

   5. *-commutative N/A

      \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{\left(z - 1\right) \cdot \frac{-1}{2}} + -1 \cdot \frac{z - 1}{y}\right) - t \]

   6. associate-*r/ N/A

      \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \frac{-1}{2} + \color{blue}{\frac{-1 \cdot \left(z - 1\right)}{y}}\right) - t \]

   7. *-commutative N/A

      \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \frac{-1}{2} + \frac{\color{blue}{\left(z - 1\right) \cdot -1}}{y}\right) - t \]

   8. associate-/l* N/A

      \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \frac{-1}{2} + \color{blue}{\left(z - 1\right) \cdot \frac{-1}{y}}\right) - t \]

   9. distribute-lft-out N/A

      \[\leadsto \left(y \cdot y\right) \cdot \color{blue}{\left(\left(z - 1\right) \cdot \left(\frac{-1}{2} + \frac{-1}{y}\right)\right)} - t \]

   10. metadata-eval N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \frac{-1}{y}\right)\right) - t \]

   11. metadata-eval N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{y}\right)\right) - t \]

   12. distribute-neg-frac N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{y}\right)\right)}\right)\right) - t \]

   13. distribute-neg-in N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} + \frac{1}{y}\right)\right)\right)}\right) - t \]

   14. mul-1-neg N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \color{blue}{\left(-1 \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)}\right) - t \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \left(y \cdot y\right) \cdot \color{blue}{\left(\left(z - 1\right) \cdot \left(-1 \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)\right)} - t \]

   16. sub-neg N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{\left(z + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(-1 \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)\right) - t \]

   17. metadata-eval N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z + \color{blue}{-1}\right) \cdot \left(-1 \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)\right) - t \]

   18. +-commutative N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{\left(-1 + z\right)} \cdot \left(-1 \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)\right) - t \]

   19. +-lowering-+.f64 N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{\left(-1 + z\right)} \cdot \left(-1 \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)\right) - t \]

   20. mul-1-neg N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} + \frac{1}{y}\right)\right)\right)}\right) - t \]

   21. distribute-neg-in N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{y}\right)\right)\right)}\right) - t \]

   22. metadata-eval N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \left(\color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(\frac{1}{y}\right)\right)\right)\right) - t \]

   23. distribute-neg-frac N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \left(\frac{-1}{2} + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{y}}\right)\right) - t \]

   24. metadata-eval N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \left(\frac{-1}{2} + \frac{\color{blue}{-1}}{y}\right)\right) - t \]

   25. +-lowering-+.f64 N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \color{blue}{\left(\frac{-1}{2} + \frac{-1}{y}\right)}\right) - t \]

   26. /-lowering-/.f64 32.4

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \left(-0.5 + \color{blue}{\frac{-1}{y}}\right)\right) - t \]

8. Simplified 32.4%

   \[\leadsto \color{blue}{\left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \left(-0.5 + \frac{-1}{y}\right)\right)} - t \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(y \cdot y\right) \cdot \left(-1 + z\right)\right) \cdot \left(\frac{-1}{2} + \frac{-1}{y}\right)} - t \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{2} + \frac{-1}{y}\right) \cdot \left(\left(y \cdot y\right) \cdot \left(-1 + z\right)\right)} - t \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{2} + \frac{-1}{y}\right) \cdot \left(\left(y \cdot y\right) \cdot \left(-1 + z\right)\right)} - t \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{2} + \frac{-1}{y}\right)} \cdot \left(\left(y \cdot y\right) \cdot \left(-1 + z\right)\right) - t \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \left(\frac{-1}{2} + \color{blue}{\frac{-1}{y}}\right) \cdot \left(\left(y \cdot y\right) \cdot \left(-1 + z\right)\right) - t \]

   6. associate-*l* N/A

      \[\leadsto \left(\frac{-1}{2} + \frac{-1}{y}\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(-1 + z\right)\right)\right)} - t \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \left(\frac{-1}{2} + \frac{-1}{y}\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(-1 + z\right)\right)\right)} - t \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \left(\frac{-1}{2} + \frac{-1}{y}\right) \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(-1 + z\right)\right)}\right) - t \]

   9. +-lowering-+.f64 48.7

      \[\leadsto \left(-0.5 + \frac{-1}{y}\right) \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(-1 + z\right)}\right)\right) - t \]

10. Applied egg-rr 48.7%

    \[\leadsto \color{blue}{\left(-0.5 + \frac{-1}{y}\right) \cdot \left(y \cdot \left(y \cdot \left(-1 + z\right)\right)\right)} - t \]
11. Add Preprocessing

Alternative 14: 46.6% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), -0.5\right), -1\right) \cdot \left(y \cdot z\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (* (fma y (fma y (fma y -0.25 -0.3333333333333333) -0.5) -1.0) (* y z)) t))

C

double code(double x, double y, double z, double t) {
	return (fma(y, fma(y, fma(y, -0.25, -0.3333333333333333), -0.5), -1.0) * (y * z)) - t;
}

Julia

function code(x, y, z, t)
	return Float64(Float64(fma(y, fma(y, fma(y, -0.25, -0.3333333333333333), -0.5), -1.0) * Float64(y * z)) - t)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(y * N[(y * N[(y * -0.25 + -0.3333333333333333), $MachinePrecision] + -0.5), $MachinePrecision] + -1.0), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 86.5%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{y \cdot \left(-1 \cdot \left(z - 1\right) + y \cdot \left(\frac{-1}{2} \cdot \left(z - 1\right) + y \cdot \left(\frac{-1}{3} \cdot \left(z - 1\right) + \frac{-1}{4} \cdot \left(y \cdot \left(z - 1\right)\right)\right)\right)\right)}\right) - t \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{y \cdot \left(-1 \cdot \left(z - 1\right) + y \cdot \left(\frac{-1}{2} \cdot \left(z - 1\right) + y \cdot \left(\frac{-1}{3} \cdot \left(z - 1\right) + \frac{-1}{4} \cdot \left(y \cdot \left(z - 1\right)\right)\right)\right)\right)}\right) - t \]

   2. +-commutative N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + y \cdot \color{blue}{\left(y \cdot \left(\frac{-1}{2} \cdot \left(z - 1\right) + y \cdot \left(\frac{-1}{3} \cdot \left(z - 1\right) + \frac{-1}{4} \cdot \left(y \cdot \left(z - 1\right)\right)\right)\right) + -1 \cdot \left(z - 1\right)\right)}\right) - t \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + y \cdot \color{blue}{\mathsf{fma}\left(y, \frac{-1}{2} \cdot \left(z - 1\right) + y \cdot \left(\frac{-1}{3} \cdot \left(z - 1\right) + \frac{-1}{4} \cdot \left(y \cdot \left(z - 1\right)\right)\right), -1 \cdot \left(z - 1\right)\right)}\right) - t \]

5. Simplified 99.8%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \left(-1 + z\right) \cdot \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), \mathsf{fma}\left(z, -0.5, 0.5\right)\right), 1 - z\right)}\right) - t \]

6. Taylor expanded in z around inf

   \[\leadsto \color{blue}{y \cdot \left(z \cdot \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} - t \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto y \cdot \color{blue}{\left(\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot z\right)} - t \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right) \cdot z} - t \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot y\right)} \cdot z - t \]

   4. associate-*l* N/A

      \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot \left(y \cdot z\right)} - t \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot \left(y \cdot z\right)} - t \]

   6. sub-neg N/A

      \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(y \cdot z\right) - t \]

   7. metadata-eval N/A

      \[\leadsto \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right) \cdot \left(y \cdot z\right) - t \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}, -1\right)} \cdot \left(y \cdot z\right) - t \]

   9. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right) \cdot \left(y \cdot z\right) - t \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(y, y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right) \cdot \left(y \cdot z\right) - t \]

   11. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{-1}{4} \cdot y - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right) \cdot \left(y \cdot z\right) - t \]

   12. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\frac{-1}{4} \cdot y + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right) \cdot \left(y \cdot z\right) - t \]

   13. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right) \cdot \left(y \cdot z\right) - t \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, y \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right) \cdot \left(y \cdot z\right) - t \]

   15. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{-1}{4}, \frac{-1}{3}\right)}, \frac{-1}{2}\right), -1\right) \cdot \left(y \cdot z\right) - t \]

   16. *-lowering-*.f64 48.7

       \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), -0.5\right), -1\right) \cdot \color{blue}{\left(y \cdot z\right)} - t \]

8. Simplified 48.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), -0.5\right), -1\right) \cdot \left(y \cdot z\right)} - t \]
9. Add Preprocessing

Alternative 15: 43.4% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{+21}:\\ \;\;\;\;0 - t\\ \mathbf{elif}\;t \leq 12500000:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;0 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2.6e+21)
   (- 0.0 t)
   (if (<= t 12500000.0) (* y (- 1.0 z)) (- 0.0 t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.6e+21) {
		tmp = 0.0 - t;
	} else if (t <= 12500000.0) {
		tmp = y * (1.0 - z);
	} else {
		tmp = 0.0 - 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 <= (-2.6d+21)) then
        tmp = 0.0d0 - t
    else if (t <= 12500000.0d0) then
        tmp = y * (1.0d0 - z)
    else
        tmp = 0.0d0 - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.6e+21) {
		tmp = 0.0 - t;
	} else if (t <= 12500000.0) {
		tmp = y * (1.0 - z);
	} else {
		tmp = 0.0 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -2.6e+21:
		tmp = 0.0 - t
	elif t <= 12500000.0:
		tmp = y * (1.0 - z)
	else:
		tmp = 0.0 - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2.6e+21)
		tmp = Float64(0.0 - t);
	elseif (t <= 12500000.0)
		tmp = Float64(y * Float64(1.0 - z));
	else
		tmp = Float64(0.0 - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -2.6e+21)
		tmp = 0.0 - t;
	elseif (t <= 12500000.0)
		tmp = y * (1.0 - z);
	else
		tmp = 0.0 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2.6e+21], N[(0.0 - t), $MachinePrecision], If[LessEqual[t, 12500000.0], N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(0.0 - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.6e21 or 1.25e7 < t

   1. Initial program 95.9%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in 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 76.5

         \[\leadsto \color{blue}{0 - t} \]

   5. Simplified 76.5%

      \[\leadsto \color{blue}{0 - t} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(t\right)} \]

      2. neg-lowering-neg.f64 76.5

         \[\leadsto \color{blue}{-t} \]

   7. Applied egg-rr 76.5%

      \[\leadsto \color{blue}{-t} \]

   if -2.6e21 < t < 1.25e7

   1. Initial program 79.1%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + \log y \cdot \left(x - 1\right)\right) - t} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\log y \cdot \left(x - 1\right) + -1 \cdot \left(y \cdot \left(z - 1\right)\right)\right)} - t \]

      2. mul-1-neg N/A

         \[\leadsto \left(\log y \cdot \left(x - 1\right) + \color{blue}{\left(\mathsf{neg}\left(y \cdot \left(z - 1\right)\right)\right)}\right) - t \]

      3. unsub-neg N/A

         \[\leadsto \color{blue}{\left(\log y \cdot \left(x - 1\right) - y \cdot \left(z - 1\right)\right)} - t \]

      4. associate--l- N/A

         \[\leadsto \color{blue}{\log y \cdot \left(x - 1\right) - \left(y \cdot \left(z - 1\right) + t\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\log y \cdot \left(x - 1\right) - \left(y \cdot \left(z - 1\right) + t\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\log y \cdot \left(x - 1\right)} - \left(y \cdot \left(z - 1\right) + t\right) \]

      7. log-lowering-log.f64 N/A

         \[\leadsto \color{blue}{\log y} \cdot \left(x - 1\right) - \left(y \cdot \left(z - 1\right) + t\right) \]

      8. sub-neg N/A

         \[\leadsto \log y \cdot \color{blue}{\left(x + \left(\mathsf{neg}\left(1\right)\right)\right)} - \left(y \cdot \left(z - 1\right) + t\right) \]

      9. metadata-eval N/A

         \[\leadsto \log y \cdot \left(x + \color{blue}{-1}\right) - \left(y \cdot \left(z - 1\right) + t\right) \]

      10. +-commutative N/A

          \[\leadsto \log y \cdot \color{blue}{\left(-1 + x\right)} - \left(y \cdot \left(z - 1\right) + t\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \log y \cdot \color{blue}{\left(-1 + x\right)} - \left(y \cdot \left(z - 1\right) + t\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \log y \cdot \left(-1 + x\right) - \color{blue}{\mathsf{fma}\left(y, z - 1, t\right)} \]

      13. sub-neg N/A

          \[\leadsto \log y \cdot \left(-1 + x\right) - \mathsf{fma}\left(y, \color{blue}{z + \left(\mathsf{neg}\left(1\right)\right)}, t\right) \]

      14. metadata-eval N/A

          \[\leadsto \log y \cdot \left(-1 + x\right) - \mathsf{fma}\left(y, z + \color{blue}{-1}, t\right) \]

      15. +-commutative N/A

          \[\leadsto \log y \cdot \left(-1 + x\right) - \mathsf{fma}\left(y, \color{blue}{-1 + z}, t\right) \]

      16. +-lowering-+.f64 99.4

          \[\leadsto \log y \cdot \left(-1 + x\right) - \mathsf{fma}\left(y, \color{blue}{-1 + z}, t\right) \]

   5. Simplified 99.4%

      \[\leadsto \color{blue}{\log y \cdot \left(-1 + x\right) - \mathsf{fma}\left(y, -1 + z, t\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(1 - z\right)} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto y \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)} \]

      2. neg-mul-1 N/A

         \[\leadsto y \cdot \left(1 + \color{blue}{-1 \cdot z}\right) \]

      3. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(-1 \cdot z + 1\right)} \]

      4. neg-mul-1 N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + 1\right) \]

      5. neg-sub0 N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(0 - z\right)} + 1\right) \]

      6. associate-+l- N/A

         \[\leadsto y \cdot \color{blue}{\left(0 - \left(z - 1\right)\right)} \]

      7. neg-sub0 N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(z - 1\right)\right)\right)} \]

      8. mul-1-neg N/A

         \[\leadsto y \cdot \color{blue}{\left(-1 \cdot \left(z - 1\right)\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(z - 1\right)\right)} \]

      10. mul-1-neg N/A

          \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(z - 1\right)\right)\right)} \]

      11. neg-sub0 N/A

          \[\leadsto y \cdot \color{blue}{\left(0 - \left(z - 1\right)\right)} \]

      12. sub-neg N/A

          \[\leadsto y \cdot \left(0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

      13. metadata-eval N/A

          \[\leadsto y \cdot \left(0 - \left(z + \color{blue}{-1}\right)\right) \]

      14. +-commutative N/A

          \[\leadsto y \cdot \left(0 - \color{blue}{\left(-1 + z\right)}\right) \]

      15. associate--r+ N/A

          \[\leadsto y \cdot \color{blue}{\left(\left(0 - -1\right) - z\right)} \]

      16. metadata-eval N/A

          \[\leadsto y \cdot \left(\color{blue}{1} - z\right) \]

      17. --lowering--.f64 23.2

          \[\leadsto y \cdot \color{blue}{\left(1 - z\right)} \]

   8. Simplified 23.2%

      \[\leadsto \color{blue}{y \cdot \left(1 - z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{+21}:\\ \;\;\;\;0 - t\\ \mathbf{elif}\;t \leq 12500000:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;0 - t\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 46.5% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ z \cdot \left(y \cdot \mathsf{fma}\left(y, -0.5, -1\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* z (* y (fma y -0.5 -1.0))) t))

C

double code(double x, double y, double z, double t) {
	return (z * (y * fma(y, -0.5, -1.0))) - t;
}

Julia

function code(x, y, z, t)
	return Float64(Float64(z * Float64(y * fma(y, -0.5, -1.0))) - t)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(z * N[(y * N[(y * -0.5 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 86.5%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(y \cdot \left(-1 \cdot \left(z - 1\right) + \frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(-1 \cdot \left(z - 1\right) + \frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right)\right) \cdot y} + \log y \cdot \left(x - 1\right)\right) - t \]

   2. associate-*r* N/A

      \[\leadsto \left(\left(-1 \cdot \left(z - 1\right) + \color{blue}{\left(\frac{-1}{2} \cdot y\right) \cdot \left(z - 1\right)}\right) \cdot y + \log y \cdot \left(x - 1\right)\right) - t \]

   3. distribute-rgt-out N/A

      \[\leadsto \left(\color{blue}{\left(\left(z - 1\right) \cdot \left(-1 + \frac{-1}{2} \cdot y\right)\right)} \cdot y + \log y \cdot \left(x - 1\right)\right) - t \]

   4. +-commutative N/A

      \[\leadsto \left(\left(\left(z - 1\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot y + -1\right)}\right) \cdot y + \log y \cdot \left(x - 1\right)\right) - t \]

   5. metadata-eval N/A

      \[\leadsto \left(\left(\left(z - 1\right) \cdot \left(\frac{-1}{2} \cdot y + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \cdot y + \log y \cdot \left(x - 1\right)\right) - t \]

   6. sub-neg N/A

      \[\leadsto \left(\left(\left(z - 1\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot y - 1\right)}\right) \cdot y + \log y \cdot \left(x - 1\right)\right) - t \]

   7. associate-*l* N/A

      \[\leadsto \left(\color{blue}{\left(z - 1\right) \cdot \left(\left(\frac{-1}{2} \cdot y - 1\right) \cdot y\right)} + \log y \cdot \left(x - 1\right)\right) - t \]

   8. *-commutative N/A

      \[\leadsto \left(\left(z - 1\right) \cdot \color{blue}{\left(y \cdot \left(\frac{-1}{2} \cdot y - 1\right)\right)} + \log y \cdot \left(x - 1\right)\right) - t \]

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(z - 1, y \cdot \left(\frac{-1}{2} \cdot y - 1\right), \log y \cdot \left(x - 1\right)\right)} - t \]

   10. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{z + \left(\mathsf{neg}\left(1\right)\right)}, y \cdot \left(\frac{-1}{2} \cdot y - 1\right), \log y \cdot \left(x - 1\right)\right) - t \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(z + \color{blue}{-1}, y \cdot \left(\frac{-1}{2} \cdot y - 1\right), \log y \cdot \left(x - 1\right)\right) - t \]

   12. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{-1 + z}, y \cdot \left(\frac{-1}{2} \cdot y - 1\right), \log y \cdot \left(x - 1\right)\right) - t \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{-1 + z}, y \cdot \left(\frac{-1}{2} \cdot y - 1\right), \log y \cdot \left(x - 1\right)\right) - t \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, \color{blue}{y \cdot \left(\frac{-1}{2} \cdot y - 1\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

   15. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \color{blue}{\left(\frac{-1}{2} \cdot y + \left(\mathsf{neg}\left(1\right)\right)\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

   16. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \left(\color{blue}{y \cdot \frac{-1}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right), \log y \cdot \left(x - 1\right)\right) - t \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \left(y \cdot \frac{-1}{2} + \color{blue}{-1}\right), \log y \cdot \left(x - 1\right)\right) - t \]

   18. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \color{blue}{\mathsf{fma}\left(y, \frac{-1}{2}, -1\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

   19. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \mathsf{fma}\left(y, \frac{-1}{2}, -1\right), \color{blue}{\log y \cdot \left(x - 1\right)}\right) - t \]

   20. log-lowering-log.f64 N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \mathsf{fma}\left(y, \frac{-1}{2}, -1\right), \color{blue}{\log y} \cdot \left(x - 1\right)\right) - t \]

   21. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \mathsf{fma}\left(y, \frac{-1}{2}, -1\right), \log y \cdot \color{blue}{\left(x + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) - t \]

5. Simplified 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-1 + z, y \cdot \mathsf{fma}\left(y, -0.5, -1\right), \log y \cdot \left(-1 + x\right)\right)} - t \]

6. Taylor expanded in z around inf

   \[\leadsto \color{blue}{y \cdot \left(z \cdot \left(\frac{-1}{2} \cdot y - 1\right)\right)} - t \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(z \cdot \left(\frac{-1}{2} \cdot y - 1\right)\right) \cdot y} - t \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{z \cdot \left(\left(\frac{-1}{2} \cdot y - 1\right) \cdot y\right)} - t \]

   3. *-commutative N/A

      \[\leadsto z \cdot \color{blue}{\left(y \cdot \left(\frac{-1}{2} \cdot y - 1\right)\right)} - t \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{z \cdot \left(y \cdot \left(\frac{-1}{2} \cdot y - 1\right)\right)} - t \]

   5. *-lowering-*.f64 N/A

      \[\leadsto z \cdot \color{blue}{\left(y \cdot \left(\frac{-1}{2} \cdot y - 1\right)\right)} - t \]

   6. sub-neg N/A

      \[\leadsto z \cdot \left(y \cdot \color{blue}{\left(\frac{-1}{2} \cdot y + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) - t \]

   7. *-commutative N/A

      \[\leadsto z \cdot \left(y \cdot \left(\color{blue}{y \cdot \frac{-1}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) - t \]

   8. metadata-eval N/A

      \[\leadsto z \cdot \left(y \cdot \left(y \cdot \frac{-1}{2} + \color{blue}{-1}\right)\right) - t \]

   9. accelerator-lowering-fma.f64 48.6

      \[\leadsto z \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(y, -0.5, -1\right)}\right) - t \]

8. Simplified 48.6%

   \[\leadsto \color{blue}{z \cdot \left(y \cdot \mathsf{fma}\left(y, -0.5, -1\right)\right)} - t \]
9. Add Preprocessing

Alternative 17: 46.4% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ y - \mathsf{fma}\left(y, z, t\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- y (fma y z t)))

C

double code(double x, double y, double z, double t) {
	return y - fma(y, z, t);
}

Julia

function code(x, y, z, t)
	return Float64(y - fma(y, z, t))
end

Wolfram

code[x_, y_, z_, t_] := N[(y - N[(y * z + t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.5%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(y \cdot \left(-1 \cdot \left(z - 1\right) + \frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(-1 \cdot \left(z - 1\right) + \frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right)\right) \cdot y} + \log y \cdot \left(x - 1\right)\right) - t \]

   2. associate-*r* N/A

      \[\leadsto \left(\left(-1 \cdot \left(z - 1\right) + \color{blue}{\left(\frac{-1}{2} \cdot y\right) \cdot \left(z - 1\right)}\right) \cdot y + \log y \cdot \left(x - 1\right)\right) - t \]

   3. distribute-rgt-out N/A

      \[\leadsto \left(\color{blue}{\left(\left(z - 1\right) \cdot \left(-1 + \frac{-1}{2} \cdot y\right)\right)} \cdot y + \log y \cdot \left(x - 1\right)\right) - t \]

   4. +-commutative N/A

      \[\leadsto \left(\left(\left(z - 1\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot y + -1\right)}\right) \cdot y + \log y \cdot \left(x - 1\right)\right) - t \]

   5. metadata-eval N/A

      \[\leadsto \left(\left(\left(z - 1\right) \cdot \left(\frac{-1}{2} \cdot y + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \cdot y + \log y \cdot \left(x - 1\right)\right) - t \]

   6. sub-neg N/A

      \[\leadsto \left(\left(\left(z - 1\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot y - 1\right)}\right) \cdot y + \log y \cdot \left(x - 1\right)\right) - t \]

   7. associate-*l* N/A

      \[\leadsto \left(\color{blue}{\left(z - 1\right) \cdot \left(\left(\frac{-1}{2} \cdot y - 1\right) \cdot y\right)} + \log y \cdot \left(x - 1\right)\right) - t \]

   8. *-commutative N/A

      \[\leadsto \left(\left(z - 1\right) \cdot \color{blue}{\left(y \cdot \left(\frac{-1}{2} \cdot y - 1\right)\right)} + \log y \cdot \left(x - 1\right)\right) - t \]

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(z - 1, y \cdot \left(\frac{-1}{2} \cdot y - 1\right), \log y \cdot \left(x - 1\right)\right)} - t \]

   10. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{z + \left(\mathsf{neg}\left(1\right)\right)}, y \cdot \left(\frac{-1}{2} \cdot y - 1\right), \log y \cdot \left(x - 1\right)\right) - t \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(z + \color{blue}{-1}, y \cdot \left(\frac{-1}{2} \cdot y - 1\right), \log y \cdot \left(x - 1\right)\right) - t \]

   12. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{-1 + z}, y \cdot \left(\frac{-1}{2} \cdot y - 1\right), \log y \cdot \left(x - 1\right)\right) - t \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{-1 + z}, y \cdot \left(\frac{-1}{2} \cdot y - 1\right), \log y \cdot \left(x - 1\right)\right) - t \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, \color{blue}{y \cdot \left(\frac{-1}{2} \cdot y - 1\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

   15. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \color{blue}{\left(\frac{-1}{2} \cdot y + \left(\mathsf{neg}\left(1\right)\right)\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

   16. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \left(\color{blue}{y \cdot \frac{-1}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right), \log y \cdot \left(x - 1\right)\right) - t \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \left(y \cdot \frac{-1}{2} + \color{blue}{-1}\right), \log y \cdot \left(x - 1\right)\right) - t \]

   18. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \color{blue}{\mathsf{fma}\left(y, \frac{-1}{2}, -1\right)}, \log y \cdot \left(x - 1\right)\right) - t \]

   19. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \mathsf{fma}\left(y, \frac{-1}{2}, -1\right), \color{blue}{\log y \cdot \left(x - 1\right)}\right) - t \]

   20. log-lowering-log.f64 N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \mathsf{fma}\left(y, \frac{-1}{2}, -1\right), \color{blue}{\log y} \cdot \left(x - 1\right)\right) - t \]

   21. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(-1 + z, y \cdot \mathsf{fma}\left(y, \frac{-1}{2}, -1\right), \log y \cdot \color{blue}{\left(x + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) - t \]

5. Simplified 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-1 + z, y \cdot \mathsf{fma}\left(y, -0.5, -1\right), \log y \cdot \left(-1 + x\right)\right)} - t \]

6. Taylor expanded in y around inf

   \[\leadsto \color{blue}{{y}^{2} \cdot \left(-1 \cdot \frac{z - 1}{y} + \frac{-1}{2} \cdot \left(z - 1\right)\right)} - t \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{{y}^{2} \cdot \left(-1 \cdot \frac{z - 1}{y} + \frac{-1}{2} \cdot \left(z - 1\right)\right)} - t \]

   2. unpow2 N/A

      \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot \left(-1 \cdot \frac{z - 1}{y} + \frac{-1}{2} \cdot \left(z - 1\right)\right) - t \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot \left(-1 \cdot \frac{z - 1}{y} + \frac{-1}{2} \cdot \left(z - 1\right)\right) - t \]

   4. +-commutative N/A

      \[\leadsto \left(y \cdot y\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \left(z - 1\right) + -1 \cdot \frac{z - 1}{y}\right)} - t \]

   5. *-commutative N/A

      \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{\left(z - 1\right) \cdot \frac{-1}{2}} + -1 \cdot \frac{z - 1}{y}\right) - t \]

   6. associate-*r/ N/A

      \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \frac{-1}{2} + \color{blue}{\frac{-1 \cdot \left(z - 1\right)}{y}}\right) - t \]

   7. *-commutative N/A

      \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \frac{-1}{2} + \frac{\color{blue}{\left(z - 1\right) \cdot -1}}{y}\right) - t \]

   8. associate-/l* N/A

      \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \frac{-1}{2} + \color{blue}{\left(z - 1\right) \cdot \frac{-1}{y}}\right) - t \]

   9. distribute-lft-out N/A

      \[\leadsto \left(y \cdot y\right) \cdot \color{blue}{\left(\left(z - 1\right) \cdot \left(\frac{-1}{2} + \frac{-1}{y}\right)\right)} - t \]

   10. metadata-eval N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \frac{-1}{y}\right)\right) - t \]

   11. metadata-eval N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{y}\right)\right) - t \]

   12. distribute-neg-frac N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{y}\right)\right)}\right)\right) - t \]

   13. distribute-neg-in N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} + \frac{1}{y}\right)\right)\right)}\right) - t \]

   14. mul-1-neg N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z - 1\right) \cdot \color{blue}{\left(-1 \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)}\right) - t \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \left(y \cdot y\right) \cdot \color{blue}{\left(\left(z - 1\right) \cdot \left(-1 \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)\right)} - t \]

   16. sub-neg N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{\left(z + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(-1 \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)\right) - t \]

   17. metadata-eval N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(z + \color{blue}{-1}\right) \cdot \left(-1 \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)\right) - t \]

   18. +-commutative N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{\left(-1 + z\right)} \cdot \left(-1 \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)\right) - t \]

   19. +-lowering-+.f64 N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{\left(-1 + z\right)} \cdot \left(-1 \cdot \left(\frac{1}{2} + \frac{1}{y}\right)\right)\right) - t \]

   20. mul-1-neg N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} + \frac{1}{y}\right)\right)\right)}\right) - t \]

   21. distribute-neg-in N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{y}\right)\right)\right)}\right) - t \]

   22. metadata-eval N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \left(\color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(\frac{1}{y}\right)\right)\right)\right) - t \]

   23. distribute-neg-frac N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \left(\frac{-1}{2} + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{y}}\right)\right) - t \]

   24. metadata-eval N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \left(\frac{-1}{2} + \frac{\color{blue}{-1}}{y}\right)\right) - t \]

   25. +-lowering-+.f64 N/A

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \color{blue}{\left(\frac{-1}{2} + \frac{-1}{y}\right)}\right) - t \]

   26. /-lowering-/.f64 32.4

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \left(-0.5 + \color{blue}{\frac{-1}{y}}\right)\right) - t \]

8. Simplified 32.4%

   \[\leadsto \color{blue}{\left(y \cdot y\right) \cdot \left(\left(-1 + z\right) \cdot \left(-0.5 + \frac{-1}{y}\right)\right)} - t \]

9. Taylor expanded in y around 0

   \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(z - 1\right)\right) - t} \]
10. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot \left(z - 1\right)\right)\right)} - t \]

    2. neg-sub0 N/A

       \[\leadsto \color{blue}{\left(0 - y \cdot \left(z - 1\right)\right)} - t \]

    3. associate--r+ N/A

       \[\leadsto \color{blue}{0 - \left(y \cdot \left(z - 1\right) + t\right)} \]

    4. sub-neg N/A

       \[\leadsto 0 - \left(y \cdot \color{blue}{\left(z + \left(\mathsf{neg}\left(1\right)\right)\right)} + t\right) \]

    5. metadata-eval N/A

       \[\leadsto 0 - \left(y \cdot \left(z + \color{blue}{-1}\right) + t\right) \]

    6. distribute-lft-in N/A

       \[\leadsto 0 - \left(\color{blue}{\left(y \cdot z + y \cdot -1\right)} + t\right) \]

    7. *-commutative N/A

       \[\leadsto 0 - \left(\left(y \cdot z + \color{blue}{-1 \cdot y}\right) + t\right) \]

    8. associate-+l+ N/A

       \[\leadsto 0 - \color{blue}{\left(y \cdot z + \left(-1 \cdot y + t\right)\right)} \]

    9. +-commutative N/A

       \[\leadsto 0 - \left(y \cdot z + \color{blue}{\left(t + -1 \cdot y\right)}\right) \]

    10. +-commutative N/A

        \[\leadsto 0 - \color{blue}{\left(\left(t + -1 \cdot y\right) + y \cdot z\right)} \]

    11. associate-+r+ N/A

        \[\leadsto 0 - \color{blue}{\left(t + \left(-1 \cdot y + y \cdot z\right)\right)} \]

    12. +-commutative N/A

        \[\leadsto 0 - \color{blue}{\left(\left(-1 \cdot y + y \cdot z\right) + t\right)} \]

    13. associate-+l+ N/A

        \[\leadsto 0 - \color{blue}{\left(-1 \cdot y + \left(y \cdot z + t\right)\right)} \]

    14. associate--r+ N/A

        \[\leadsto \color{blue}{\left(0 - -1 \cdot y\right) - \left(y \cdot z + t\right)} \]

    15. neg-sub0 N/A

        \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-1 \cdot y\right)\right)} - \left(y \cdot z + t\right) \]

    16. mul-1-neg N/A

        \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) - \left(y \cdot z + t\right) \]

    17. remove-double-neg N/A

        \[\leadsto \color{blue}{y} - \left(y \cdot z + t\right) \]

    18. --lowering--.f64 N/A

        \[\leadsto \color{blue}{y - \left(y \cdot z + t\right)} \]

    19. accelerator-lowering-fma.f64 48.6

        \[\leadsto y - \color{blue}{\mathsf{fma}\left(y, z, t\right)} \]

11. Simplified 48.6%

    \[\leadsto \color{blue}{y - \mathsf{fma}\left(y, z, t\right)} \]
12. Add Preprocessing

Alternative 18: 36.0% accurate, 56.5× 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 86.5%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in 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 35.4

      \[\leadsto \color{blue}{0 - t} \]

5. Simplified 35.4%

   \[\leadsto \color{blue}{0 - t} \]
6. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(t\right)} \]

   2. neg-lowering-neg.f64 35.4

      \[\leadsto \color{blue}{-t} \]

7. Applied egg-rr 35.4%

   \[\leadsto \color{blue}{-t} \]

8. Final simplification 35.4%

   \[\leadsto 0 - t \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024195 
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  :precision binary64
  (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))