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

Percentage Accurate: 85.4% → 99.8%

Time: 18.9s

Alternatives: 19

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \log y + z \cdot \mathsf{log1p}\left(0 - y\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (+ (* x (log y)) (* z (log1p (- 0.0 y)))) t))

C

double code(double x, double y, double z, double t) {
	return ((x * log(y)) + (z * log1p((0.0 - y)))) - t;
}

Java

public static double code(double x, double y, double z, double t) {
	return ((x * Math.log(y)) + (z * Math.log1p((0.0 - y)))) - t;
}

Python

def code(x, y, z, t):
	return ((x * math.log(y)) + (z * math.log1p((0.0 - y)))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * log(y)) + Float64(z * log1p(Float64(0.0 - y)))) - t)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Log[1 + N[(0.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 83.8%

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

   1. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \log \left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), t\right) \]

   2. accelerator-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), t\right) \]

   3. neg-sub0 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{log1p.f64}\left(\left(0 - y\right)\right)\right)\right), t\right) \]

   4. --lowering--.f64 99.8%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{log1p.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right)\right)\right), t\right) \]

4. Applied egg-rr 99.8%

   \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\mathsf{log1p}\left(0 - y\right)}\right) - t \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

3. Taylor expanded in y around 0

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \color{blue}{\left(y \cdot \left(-1 \cdot z + y \cdot \left(\frac{-1}{2} \cdot z + y \cdot \left(\frac{-1}{3} \cdot z + \frac{-1}{4} \cdot \left(y \cdot z\right)\right)\right)\right)\right)}\right), t\right) \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(y, \left(-1 \cdot z + y \cdot \left(\frac{-1}{2} \cdot z + y \cdot \left(\frac{-1}{3} \cdot z + \frac{-1}{4} \cdot \left(y \cdot z\right)\right)\right)\right)\right)\right), t\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{2} \cdot z + y \cdot \left(\frac{-1}{3} \cdot z + \frac{-1}{4} \cdot \left(y \cdot z\right)\right)\right) + -1 \cdot z\right)\right)\right), t\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \left(\frac{-1}{2} \cdot z + y \cdot \left(\frac{-1}{3} \cdot z + \frac{-1}{4} \cdot \left(y \cdot z\right)\right)\right), \left(-1 \cdot z\right)\right)\right)\right), t\right) \]

5. Simplified 99.7%

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

Alternative 3: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

3. Taylor expanded in y around 0

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}\right)\right), t\right) \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)\right)\right), t\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right), t\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) + -1\right)\right)\right)\right), t\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right), -1\right)\right)\right)\right), t\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), -1\right)\right)\right)\right), t\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) + \frac{-1}{2}\right), -1\right)\right)\right)\right), t\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(y, \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right), \frac{-1}{2}\right), -1\right)\right)\right)\right), t\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(y, \left(\frac{-1}{4} \cdot y + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \frac{-1}{2}\right), -1\right)\right)\right)\right), t\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(y, \left(y \cdot \frac{-1}{4} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \frac{-1}{2}\right), -1\right)\right)\right)\right), t\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(y, \left(y \cdot \frac{-1}{4} + \frac{-1}{3}\right), \frac{-1}{2}\right), -1\right)\right)\right)\right), t\right) \]

   11. accelerator-lowering-fma.f64 99.7%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right), -1\right)\right)\right)\right), t\right) \]

5. Simplified 99.7%

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

Alternative 4: 99.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

3. Taylor expanded in y around 0

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

5. Simplified 99.6%

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

Alternative 5: 99.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

3. Taylor expanded in y around 0

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)}\right)\right), t\right) \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)\right)\right), t\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right), t\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) + -1\right)\right)\right)\right), t\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right), -1\right)\right)\right)\right), t\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \left(\frac{-1}{3} \cdot y + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), -1\right)\right)\right)\right), t\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \left(y \cdot \frac{-1}{3} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), -1\right)\right)\right)\right), t\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \left(y \cdot \frac{-1}{3} + \frac{-1}{2}\right), -1\right)\right)\right)\right), t\right) \]

   8. accelerator-lowering-fma.f64 99.6%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)\right)\right), t\right) \]

5. Simplified 99.6%

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

Alternative 6: 99.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. associate-*r* N/A

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

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

      \[\leadsto \mathsf{fma.f64}\left(\left(y \cdot z\right), \color{blue}{\left(\frac{-1}{2} \cdot y - 1\right)}, \left(x \cdot \log y - t\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma.f64}\left(\left(z \cdot y\right), \left(\color{blue}{\frac{-1}{2} \cdot y} - 1\right), \left(x \cdot \log y - t\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(z, y\right), \left(\color{blue}{\frac{-1}{2} \cdot y} - 1\right), \left(x \cdot \log y - t\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(z, y\right), \left(\frac{-1}{2} \cdot y + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right), \left(x \cdot \log y - t\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(z, y\right), \left(y \cdot \frac{-1}{2} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right), \left(x \cdot \log y - t\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(z, y\right), \left(y \cdot \frac{-1}{2} + -1\right), \left(x \cdot \log y - t\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(z, y\right), \mathsf{fma.f64}\left(y, \color{blue}{\frac{-1}{2}}, -1\right), \left(x \cdot \log y - t\right)\right) \]

   16. sub-neg N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(z, y\right), \mathsf{fma.f64}\left(y, \frac{-1}{2}, \color{blue}{-1}\right), \left(x \cdot \log y + \left(\mathsf{neg}\left(t\right)\right)\right)\right) \]

   17. remove-double-neg N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(z, y\right), \mathsf{fma.f64}\left(y, \frac{-1}{2}, -1\right), \left(x \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right) + \left(\mathsf{neg}\left(t\right)\right)\right)\right) \]

   18. mul-1-neg N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(z, y\right), \mathsf{fma.f64}\left(y, \frac{-1}{2}, -1\right), \left(x \cdot \left(\mathsf{neg}\left(-1 \cdot \log y\right)\right) + \left(\mathsf{neg}\left(t\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(z, y\right), \mathsf{fma.f64}\left(y, \frac{-1}{2}, -1\right), \left(\left(\mathsf{neg}\left(x \cdot \left(-1 \cdot \log y\right)\right)\right) + \left(\mathsf{neg}\left(t\right)\right)\right)\right) \]

5. Simplified 99.4%

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

6. Final simplification 99.4%

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

Alternative 7: 86.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - y \cdot z\\ \mathbf{if}\;z \leq -1.45 \cdot 10^{+218}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, 0 - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) (* y z))))
   (if (<= z -1.45e+218)
     t_1
     (if (<= z 1.3e+150) (fma x (log y) (- 0.0 t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - (y * z);
	double tmp;
	if (z <= -1.45e+218) {
		tmp = t_1;
	} else if (z <= 1.3e+150) {
		tmp = fma(x, log(y), (0.0 - t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - Float64(y * z))
	tmp = 0.0
	if (z <= -1.45e+218)
		tmp = t_1;
	elseif (z <= 1.3e+150)
		tmp = fma(x, log(y), Float64(0.0 - t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.45e+218], t$95$1, If[LessEqual[z, 1.3e+150], N[(x * N[Log[y], $MachinePrecision] + N[(0.0 - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.45e218 or 1.30000000000000003e150 < z

   1. Initial program 42.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

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

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

      7. neg-mul-1 N/A

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

      8. mul-1-neg N/A

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

      9. log-rec N/A

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

      10. associate--l- N/A

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

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

          \[\leadsto \mathsf{\_.f64}\left(\left(-1 \cdot \left(x \cdot \log \left(\frac{1}{y}\right)\right)\right), \color{blue}{\left(y \cdot z + t\right)}\right) \]

   5. Simplified 97.8%

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

   6. Taylor expanded in t around 0

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

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

         \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \log y\right), \color{blue}{\left(y \cdot z\right)}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \log y\right), \left(\color{blue}{y} \cdot z\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \left(y \cdot z\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \left(z \cdot \color{blue}{y}\right)\right) \]

      5. *-lowering-*.f64 88.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \color{blue}{y}\right)\right) \]

   8. Simplified 88.7%

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

   if -1.45e218 < z < 1.30000000000000003e150

   1. Initial program 94.4%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

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

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

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

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. log-rec N/A

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

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

         \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\left(-1 \cdot \log \left(\frac{1}{y}\right)\right)}, \left(\mathsf{neg}\left(t\right)\right)\right) \]

      10. log-rec N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \left(-1 \cdot \left(\mathsf{neg}\left(\log y\right)\right)\right), \left(\mathsf{neg}\left(t\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \left(-1 \cdot \left(-1 \cdot \color{blue}{\log y}\right)\right), \left(\mathsf{neg}\left(t\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \left(\mathsf{neg}\left(-1 \cdot \log y\right)\right), \left(\mathsf{neg}\left(t\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right), \left(\mathsf{neg}\left(t\right)\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \log y, \left(\mathsf{neg}\left(t\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{log.f64}\left(y\right), \left(\mathsf{neg}\left(t\right)\right)\right) \]

      16. neg-sub0 N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{log.f64}\left(y\right), \left(0 - t\right)\right) \]

      17. --lowering--.f64 93.6%

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{log.f64}\left(y\right), \mathsf{\_.f64}\left(0, t\right)\right) \]

   5. Simplified 93.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, 0 - t\right)} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{log.f64}\left(y\right), \left(\mathsf{neg}\left(t\right)\right)\right) \]

      2. neg-lowering-neg.f64 93.6%

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{log.f64}\left(y\right), \mathsf{neg.f64}\left(t\right)\right) \]

   7. Applied egg-rr 93.6%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+218}:\\ \;\;\;\;x \cdot \log y - y \cdot z\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, 0 - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-146}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-101}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), 0 - z\right), 0 - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, 0 - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.95e-146)
   (- (* x (log y)) t)
   (if (<= x 7e-101)
     (fma y (fma z (* y (fma y -0.3333333333333333 -0.5)) (- 0.0 z)) (- 0.0 t))
     (fma x (log y) (- 0.0 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.95e-146) {
		tmp = (x * log(y)) - t;
	} else if (x <= 7e-101) {
		tmp = fma(y, fma(z, (y * fma(y, -0.3333333333333333, -0.5)), (0.0 - z)), (0.0 - t));
	} else {
		tmp = fma(x, log(y), (0.0 - t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.95e-146)
		tmp = Float64(Float64(x * log(y)) - t);
	elseif (x <= 7e-101)
		tmp = fma(y, fma(z, Float64(y * fma(y, -0.3333333333333333, -0.5)), Float64(0.0 - z)), Float64(0.0 - t));
	else
		tmp = fma(x, log(y), Float64(0.0 - t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.95e-146], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 7e-101], N[(y * N[(z * N[(y * N[(y * -0.3333333333333333 + -0.5), $MachinePrecision]), $MachinePrecision] + N[(0.0 - z), $MachinePrecision]), $MachinePrecision] + N[(0.0 - t), $MachinePrecision]), $MachinePrecision], N[(x * N[Log[y], $MachinePrecision] + N[(0.0 - t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.95000000000000001e-146

   1. Initial program 90.9%

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

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \log \left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), t\right) \]

      2. accelerator-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), t\right) \]

      3. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{log1p.f64}\left(\left(0 - y\right)\right)\right)\right), t\right) \]

      4. --lowering--.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{log1p.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right)\right)\right), t\right) \]

   4. Applied egg-rr 99.8%

      \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\mathsf{log1p}\left(0 - y\right)}\right) - t \]

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(x \cdot \log y\right)}, t\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \log y\right), t\right) \]

      2. log-lowering-log.f64 89.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), t\right) \]

   7. Simplified 89.9%

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

   if -1.95000000000000001e-146 < x < 6.99999999999999989e-101

   1. Initial program 68.9%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \mathsf{\_.f64}\left(0, z\right)\right), \color{blue}{\left(-1 \cdot t\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \color{blue}{\mathsf{\_.f64}\left(0, z\right)}\right), \left(\mathsf{neg}\left(t\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \color{blue}{\mathsf{\_.f64}\left(0, z\right)}\right), \left(0 - t\right)\right) \]

      3. --lowering--.f64 93.7%

         \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \color{blue}{\mathsf{\_.f64}\left(0, z\right)}\right), \mathsf{\_.f64}\left(0, t\right)\right) \]

   8. Simplified 93.7%

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

   if 6.99999999999999989e-101 < x

   1. Initial program 93.5%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

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

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

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

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. log-rec N/A

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

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

         \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\left(-1 \cdot \log \left(\frac{1}{y}\right)\right)}, \left(\mathsf{neg}\left(t\right)\right)\right) \]

      10. log-rec N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \left(-1 \cdot \left(\mathsf{neg}\left(\log y\right)\right)\right), \left(\mathsf{neg}\left(t\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \left(-1 \cdot \left(-1 \cdot \color{blue}{\log y}\right)\right), \left(\mathsf{neg}\left(t\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \left(\mathsf{neg}\left(-1 \cdot \log y\right)\right), \left(\mathsf{neg}\left(t\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right), \left(\mathsf{neg}\left(t\right)\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \log y, \left(\mathsf{neg}\left(t\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{log.f64}\left(y\right), \left(\mathsf{neg}\left(t\right)\right)\right) \]

      16. neg-sub0 N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{log.f64}\left(y\right), \left(0 - t\right)\right) \]

      17. --lowering--.f64 92.1%

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{log.f64}\left(y\right), \mathsf{\_.f64}\left(0, t\right)\right) \]

   5. Simplified 92.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, 0 - t\right)} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{log.f64}\left(y\right), \left(\mathsf{neg}\left(t\right)\right)\right) \]

      2. neg-lowering-neg.f64 92.1%

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{log.f64}\left(y\right), \mathsf{neg.f64}\left(t\right)\right) \]

   7. Applied egg-rr 92.1%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-146}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-101}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), 0 - z\right), 0 - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, 0 - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{-146}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-101}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), 0 - z\right), 0 - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t)))
   (if (<= x -1.2e-146)
     t_1
     (if (<= x 5.4e-101)
       (fma
        y
        (fma z (* y (fma y -0.3333333333333333 -0.5)) (- 0.0 z))
        (- 0.0 t))
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - t;
	double tmp;
	if (x <= -1.2e-146) {
		tmp = t_1;
	} else if (x <= 5.4e-101) {
		tmp = fma(y, fma(z, (y * fma(y, -0.3333333333333333, -0.5)), (0.0 - z)), (0.0 - t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - t)
	tmp = 0.0
	if (x <= -1.2e-146)
		tmp = t_1;
	elseif (x <= 5.4e-101)
		tmp = fma(y, fma(z, Float64(y * fma(y, -0.3333333333333333, -0.5)), Float64(0.0 - z)), Float64(0.0 - t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[x, -1.2e-146], t$95$1, If[LessEqual[x, 5.4e-101], N[(y * N[(z * N[(y * N[(y * -0.3333333333333333 + -0.5), $MachinePrecision]), $MachinePrecision] + N[(0.0 - z), $MachinePrecision]), $MachinePrecision] + N[(0.0 - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2000000000000001e-146 or 5.4000000000000003e-101 < x

   1. Initial program 92.1%

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

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \log \left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), t\right) \]

      2. accelerator-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), t\right) \]

      3. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{log1p.f64}\left(\left(0 - y\right)\right)\right)\right), t\right) \]

      4. --lowering--.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{log1p.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right)\right)\right), t\right) \]

   4. Applied egg-rr 99.8%

      \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\mathsf{log1p}\left(0 - y\right)}\right) - t \]

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(x \cdot \log y\right)}, t\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \log y\right), t\right) \]

      2. log-lowering-log.f64 90.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), t\right) \]

   7. Simplified 90.9%

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

   if -1.2000000000000001e-146 < x < 5.4000000000000003e-101

   1. Initial program 68.9%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \mathsf{\_.f64}\left(0, z\right)\right), \color{blue}{\left(-1 \cdot t\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \color{blue}{\mathsf{\_.f64}\left(0, z\right)}\right), \left(\mathsf{neg}\left(t\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \color{blue}{\mathsf{\_.f64}\left(0, z\right)}\right), \left(0 - t\right)\right) \]

      3. --lowering--.f64 93.7%

         \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \color{blue}{\mathsf{\_.f64}\left(0, z\right)}\right), \mathsf{\_.f64}\left(0, t\right)\right) \]

   8. Simplified 93.7%

      \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), 0 - z\right), \color{blue}{0 - t}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 78.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -3.9 \cdot 10^{+49}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+53}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), 0 - z\right), 0 - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -3.9e+49)
     t_1
     (if (<= x 5e+53)
       (fma
        y
        (fma z (* y (fma y -0.3333333333333333 -0.5)) (- 0.0 z))
        (- 0.0 t))
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -3.9e+49) {
		tmp = t_1;
	} else if (x <= 5e+53) {
		tmp = fma(y, fma(z, (y * fma(y, -0.3333333333333333, -0.5)), (0.0 - z)), (0.0 - t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -3.9e+49)
		tmp = t_1;
	elseif (x <= 5e+53)
		tmp = fma(y, fma(z, Float64(y * fma(y, -0.3333333333333333, -0.5)), Float64(0.0 - z)), Float64(0.0 - t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.9e+49], t$95$1, If[LessEqual[x, 5e+53], N[(y * N[(z * N[(y * N[(y * -0.3333333333333333 + -0.5), $MachinePrecision]), $MachinePrecision] + N[(0.0 - z), $MachinePrecision]), $MachinePrecision] + N[(0.0 - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.9000000000000001e49 or 5.0000000000000004e53 < x

   1. Initial program 96.7%

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

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \log \left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), t\right) \]

      2. accelerator-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), t\right) \]

      3. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{log1p.f64}\left(\left(0 - y\right)\right)\right)\right), t\right) \]

      4. --lowering--.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{log1p.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right)\right)\right), t\right) \]

   4. Applied egg-rr 99.8%

      \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\mathsf{log1p}\left(0 - y\right)}\right) - t \]

   5. Taylor expanded in x around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\log y}\right) \]

      2. log-lowering-log.f64 82.3%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right) \]

   7. Simplified 82.3%

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

   if -3.9000000000000001e49 < x < 5.0000000000000004e53

   1. Initial program 76.3%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \mathsf{\_.f64}\left(0, z\right)\right), \color{blue}{\left(-1 \cdot t\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \color{blue}{\mathsf{\_.f64}\left(0, z\right)}\right), \left(\mathsf{neg}\left(t\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \color{blue}{\mathsf{\_.f64}\left(0, z\right)}\right), \left(0 - t\right)\right) \]

      3. --lowering--.f64 83.7%

         \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \color{blue}{\mathsf{\_.f64}\left(0, z\right)}\right), \mathsf{\_.f64}\left(0, t\right)\right) \]

   8. Simplified 83.7%

      \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), 0 - z\right), \color{blue}{0 - t}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. remove-double-neg N/A

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

   5. mul-1-neg N/A

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

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

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

   7. neg-mul-1 N/A

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

   8. mul-1-neg N/A

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

   9. log-rec N/A

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

   10. associate--l- N/A

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

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

       \[\leadsto \mathsf{\_.f64}\left(\left(-1 \cdot \left(x \cdot \log \left(\frac{1}{y}\right)\right)\right), \color{blue}{\left(y \cdot z + t\right)}\right) \]

5. Simplified 99.0%

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

Alternative 12: 57.1% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

3. Taylor expanded in y around 0

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

5. Simplified 99.6%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \mathsf{\_.f64}\left(0, z\right)\right), \color{blue}{\left(-1 \cdot t\right)}\right) \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \color{blue}{\mathsf{\_.f64}\left(0, z\right)}\right), \left(\mathsf{neg}\left(t\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \color{blue}{\mathsf{\_.f64}\left(0, z\right)}\right), \left(0 - t\right)\right) \]

   3. --lowering--.f64 59.0%

      \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \color{blue}{\mathsf{\_.f64}\left(0, z\right)}\right), \mathsf{\_.f64}\left(0, t\right)\right) \]

8. Simplified 59.0%

   \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), 0 - z\right), \color{blue}{0 - t}\right) \]
9. Step-by-step derivation

   1. distribute-rgt-in N/A

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

   2. sub0-neg N/A

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

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

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

   4. associate-+l+ N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. neg-sub0 N/A

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

   8. distribute-neg-in N/A

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

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

      \[\leadsto \mathsf{fma.f64}\left(\left(y \cdot \left(y \cdot \frac{-1}{3} + \frac{-1}{2}\right)\right), \color{blue}{\left(z \cdot y\right)}, \left(\mathsf{neg}\left(\left(z \cdot y + t\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(y, \left(y \cdot \frac{-1}{3} + \frac{-1}{2}\right)\right), \left(\color{blue}{z} \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot y + t\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \left(z \cdot y\right), \left(\mathsf{neg}\left(\left(z \cdot y + t\right)\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \left(y \cdot \color{blue}{z}\right), \left(\mathsf{neg}\left(\left(z \cdot y + t\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \mathsf{*.f64}\left(y, \color{blue}{z}\right), \left(\mathsf{neg}\left(\left(z \cdot y + t\right)\right)\right)\right) \]

   14. neg-sub0 N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \mathsf{*.f64}\left(y, z\right), \left(0 - \left(z \cdot y + t\right)\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \mathsf{*.f64}\left(y, z\right), \mathsf{\_.f64}\left(0, \left(z \cdot y + t\right)\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \mathsf{*.f64}\left(y, z\right), \mathsf{\_.f64}\left(0, \left(y \cdot z + t\right)\right)\right) \]

   17. accelerator-lowering-fma.f64 59.0%

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \mathsf{*.f64}\left(y, z\right), \mathsf{\_.f64}\left(0, \mathsf{fma.f64}\left(y, z, t\right)\right)\right) \]

10. Applied egg-rr 59.0%

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

Alternative 13: 57.1% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

3. Taylor expanded in y around 0

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

5. Simplified 99.6%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \mathsf{\_.f64}\left(0, z\right)\right), \color{blue}{\left(-1 \cdot t\right)}\right) \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \color{blue}{\mathsf{\_.f64}\left(0, z\right)}\right), \left(\mathsf{neg}\left(t\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \color{blue}{\mathsf{\_.f64}\left(0, z\right)}\right), \left(0 - t\right)\right) \]

   3. --lowering--.f64 59.0%

      \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \color{blue}{\mathsf{\_.f64}\left(0, z\right)}\right), \mathsf{\_.f64}\left(0, t\right)\right) \]

8. Simplified 59.0%

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

Alternative 14: 44.6% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(0 - z\right)\\ \mathbf{if}\;z \leq -8 \cdot 10^{+217}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.95 \cdot 10^{+155}:\\ \;\;\;\;0 - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (- 0.0 z))))
   (if (<= z -8e+217) t_1 (if (<= z 3.95e+155) (- 0.0 t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (0.0 - z);
	double tmp;
	if (z <= -8e+217) {
		tmp = t_1;
	} else if (z <= 3.95e+155) {
		tmp = 0.0 - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = y * (0.0d0 - z)
    if (z <= (-8d+217)) then
        tmp = t_1
    else if (z <= 3.95d+155) then
        tmp = 0.0d0 - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = y * (0.0 - z);
	double tmp;
	if (z <= -8e+217) {
		tmp = t_1;
	} else if (z <= 3.95e+155) {
		tmp = 0.0 - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y * (0.0 - z)
	tmp = 0
	if z <= -8e+217:
		tmp = t_1
	elif z <= 3.95e+155:
		tmp = 0.0 - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(0.0 - z))
	tmp = 0.0
	if (z <= -8e+217)
		tmp = t_1;
	elseif (z <= 3.95e+155)
		tmp = Float64(0.0 - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y * (0.0 - z);
	tmp = 0.0;
	if (z <= -8e+217)
		tmp = t_1;
	elseif (z <= 3.95e+155)
		tmp = 0.0 - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(0.0 - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -8e+217], t$95$1, If[LessEqual[z, 3.95e+155], N[(0.0 - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.99999999999999968e217 or 3.94999999999999992e155 < z

   1. Initial program 42.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

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

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

      7. neg-mul-1 N/A

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

      8. mul-1-neg N/A

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

      9. log-rec N/A

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

      10. associate--l- N/A

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

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

          \[\leadsto \mathsf{\_.f64}\left(\left(-1 \cdot \left(x \cdot \log \left(\frac{1}{y}\right)\right)\right), \color{blue}{\left(y \cdot z + t\right)}\right) \]

   5. Simplified 97.7%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(-1 \cdot y\right)}\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(y\right)\right)\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(0 - \color{blue}{y}\right)\right) \]

      8. --lowering--.f64 61.7%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(0, \color{blue}{y}\right)\right) \]

   8. Simplified 61.7%

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

   if -7.99999999999999968e217 < z < 3.94999999999999992e155

   1. Initial program 94.0%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 49.8%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{t}\right) \]

   5. Simplified 49.8%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 49.8%

         \[\leadsto \mathsf{neg.f64}\left(t\right) \]

   7. Applied egg-rr 49.8%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+217}:\\ \;\;\;\;y \cdot \left(0 - z\right)\\ \mathbf{elif}\;z \leq 3.95 \cdot 10^{+155}:\\ \;\;\;\;0 - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.0% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

3. Taylor expanded in y around 0

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

5. Simplified 99.6%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \mathsf{\_.f64}\left(0, z\right)\right), \color{blue}{\left(-1 \cdot t\right)}\right) \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \color{blue}{\mathsf{\_.f64}\left(0, z\right)}\right), \left(\mathsf{neg}\left(t\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \color{blue}{\mathsf{\_.f64}\left(0, z\right)}\right), \left(0 - t\right)\right) \]

   3. --lowering--.f64 59.0%

      \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \color{blue}{\mathsf{\_.f64}\left(0, z\right)}\right), \mathsf{\_.f64}\left(0, t\right)\right) \]

8. Simplified 59.0%

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

9. Taylor expanded in y around 0

   \[\leadsto \mathsf{fma.f64}\left(y, \color{blue}{\left(\frac{-1}{2} \cdot \left(y \cdot z\right) - z\right)}, \mathsf{\_.f64}\left(0, t\right)\right) \]
10. Step-by-step derivation

    1. sub-neg N/A

       \[\leadsto \mathsf{fma.f64}\left(y, \left(\frac{-1}{2} \cdot \left(y \cdot z\right) + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right), \mathsf{\_.f64}\left(0, t\right)\right) \]

    2. associate-*r* N/A

       \[\leadsto \mathsf{fma.f64}\left(y, \left(\left(\frac{-1}{2} \cdot y\right) \cdot z + \left(\mathsf{neg}\left(\color{blue}{z}\right)\right)\right), \mathsf{\_.f64}\left(0, t\right)\right) \]

    3. mul-1-neg N/A

       \[\leadsto \mathsf{fma.f64}\left(y, \left(\left(\frac{-1}{2} \cdot y\right) \cdot z + -1 \cdot \color{blue}{z}\right), \mathsf{\_.f64}\left(0, t\right)\right) \]

    4. distribute-rgt-out N/A

       \[\leadsto \mathsf{fma.f64}\left(y, \left(z \cdot \color{blue}{\left(\frac{-1}{2} \cdot y + -1\right)}\right), \mathsf{\_.f64}\left(0, t\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{-1}{2} \cdot y + -1\right)}\right), \mathsf{\_.f64}\left(0, t\right)\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{*.f64}\left(z, \left(y \cdot \frac{-1}{2} + -1\right)\right), \mathsf{\_.f64}\left(0, t\right)\right) \]

    7. accelerator-lowering-fma.f64 58.8%

       \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{fma.f64}\left(y, \color{blue}{\frac{-1}{2}}, -1\right)\right), \mathsf{\_.f64}\left(0, t\right)\right) \]

11. Simplified 58.8%

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

Alternative 16: 57.0% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

3. Taylor expanded in y around 0

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

5. Simplified 99.6%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \mathsf{\_.f64}\left(0, z\right)\right), \color{blue}{\left(-1 \cdot t\right)}\right) \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \color{blue}{\mathsf{\_.f64}\left(0, z\right)}\right), \left(\mathsf{neg}\left(t\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \color{blue}{\mathsf{\_.f64}\left(0, z\right)}\right), \left(0 - t\right)\right) \]

   3. --lowering--.f64 59.0%

      \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{fma.f64}\left(y, \frac{-1}{3}, \frac{-1}{2}\right)\right), \color{blue}{\mathsf{\_.f64}\left(0, z\right)}\right), \mathsf{\_.f64}\left(0, t\right)\right) \]

8. Simplified 59.0%

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

9. Taylor expanded in y around 0

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

    1. sub-neg N/A

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

    2. neg-sub0 N/A

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

    3. associate-+r- N/A

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

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

       \[\leadsto \mathsf{\_.f64}\left(\left(y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right) + 0\right), \color{blue}{t}\right) \]

    5. distribute-lft-in N/A

       \[\leadsto \mathsf{\_.f64}\left(\left(\left(y \cdot \left(-1 \cdot z\right) + y \cdot \left(\frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right) + 0\right), t\right) \]

    6. mul-1-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(\left(\left(y \cdot \left(\mathsf{neg}\left(z\right)\right) + y \cdot \left(\frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right) + 0\right), t\right) \]

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

       \[\leadsto \mathsf{\_.f64}\left(\left(\left(\left(\mathsf{neg}\left(y \cdot z\right)\right) + y \cdot \left(\frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right) + 0\right), t\right) \]

    8. mul-1-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(\left(\left(-1 \cdot \left(y \cdot z\right) + y \cdot \left(\frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right) + 0\right), t\right) \]

    9. associate-*r* N/A

       \[\leadsto \mathsf{\_.f64}\left(\left(\left(-1 \cdot \left(y \cdot z\right) + \left(y \cdot \frac{-1}{2}\right) \cdot \left(y \cdot z\right)\right) + 0\right), t\right) \]

    10. *-commutative N/A

        \[\leadsto \mathsf{\_.f64}\left(\left(\left(-1 \cdot \left(y \cdot z\right) + \left(\frac{-1}{2} \cdot y\right) \cdot \left(y \cdot z\right)\right) + 0\right), t\right) \]

    11. distribute-rgt-out N/A

        \[\leadsto \mathsf{\_.f64}\left(\left(\left(y \cdot z\right) \cdot \left(-1 + \frac{-1}{2} \cdot y\right) + 0\right), t\right) \]

    12. +-commutative N/A

        \[\leadsto \mathsf{\_.f64}\left(\left(\left(y \cdot z\right) \cdot \left(\frac{-1}{2} \cdot y + -1\right) + 0\right), t\right) \]

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

        \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\left(y \cdot z\right), \left(\frac{-1}{2} \cdot y + -1\right), 0\right), t\right) \]

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

        \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(y, z\right), \left(\frac{-1}{2} \cdot y + -1\right), 0\right), t\right) \]

    15. *-commutative N/A

        \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(y, z\right), \left(y \cdot \frac{-1}{2} + -1\right), 0\right), t\right) \]

    16. accelerator-lowering-fma.f64 58.8%

        \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{fma.f64}\left(y, \frac{-1}{2}, -1\right), 0\right), t\right) \]

11. Simplified 58.8%

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

Alternative 17: 56.7% accurate, 22.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. remove-double-neg N/A

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

   5. mul-1-neg N/A

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

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

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

   7. neg-mul-1 N/A

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

   8. mul-1-neg N/A

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

   9. log-rec N/A

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

   10. associate--l- N/A

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

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

       \[\leadsto \mathsf{\_.f64}\left(\left(-1 \cdot \left(x \cdot \log \left(\frac{1}{y}\right)\right)\right), \color{blue}{\left(y \cdot z + t\right)}\right) \]

5. Simplified 99.0%

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

6. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

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

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(t + y \cdot z\right)}\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \left(y \cdot z + \color{blue}{t}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \left(z \cdot y + t\right)\right) \]

   6. accelerator-lowering-fma.f64 58.4%

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{fma.f64}\left(z, \color{blue}{y}, t\right)\right) \]

8. Simplified 58.4%

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

Alternative 18: 42.3% accurate, 55.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 0.0d0 - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

   3. --lowering--.f64 41.9%

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{t}\right) \]

5. Simplified 41.9%

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

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 41.9%

      \[\leadsto \mathsf{neg.f64}\left(t\right) \]

7. Applied egg-rr 41.9%

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

8. Final simplification 41.9%

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

Alternative 19: 2.3% accurate, 220.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

   3. --lowering--.f64 41.9%

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{t}\right) \]

5. Simplified 41.9%

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

   1. flip3-- N/A

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

   2. metadata-eval N/A

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

   3. sub0-neg N/A

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

   4. cube-neg N/A

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

   5. sub0-neg N/A

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

   6. sqr-pow N/A

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

   7. unpow-prod-down N/A

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

   8. sub0-neg N/A

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

   9. sub0-neg N/A

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

   10. sqr-neg N/A

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

   11. unpow-prod-down N/A

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

   12. sqr-pow N/A

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

   13. metadata-eval N/A

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

   14. +-lft-identity N/A

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

   15. distribute-rgt-out N/A

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

   16. +-commutative N/A

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

   17. +-lft-identity N/A

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

   18. pow2 N/A

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

   19. pow-div N/A

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

   20. metadata-eval N/A

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

   21. unpow1 2.3%

       \[\leadsto t \]

7. Applied egg-rr 2.3%

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

Developer Target 1: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024193 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

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

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