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

Percentage Accurate: 89.6% → 99.8%

Time: 14.3s

Alternatives: 14

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z + -1, \mathsf{log1p}\left(-y\right), \left(x + -1\right) \cdot \log y - t\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma (+ z -1.0) (log1p (- y)) (- (* (+ x -1.0) (log y)) t)))

C

double code(double x, double y, double z, double t) {
	return fma((z + -1.0), log1p(-y), (((x + -1.0) * log(y)) - t));
}

Julia

function code(x, y, z, t)
	return fma(Float64(z + -1.0), log1p(Float64(-y)), Float64(Float64(Float64(x + -1.0) * log(y)) - t))
end

Wolfram

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

Derivation

1. Initial program 89.7%

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

   1. +-commutative 89.7%

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

   2. associate--l+ 89.7%

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

   3. fma-def 89.7%

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

   4. sub-neg 89.7%

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

   5. log1p-def 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x + -1\right) \cdot \log y + \left(z \cdot \left(-0.5 \cdot \left(y \cdot y\right) + {y}^{3} \cdot -0.3333333333333333\right) - y \cdot \left(z + -1\right)\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (+
   (* (+ x -1.0) (log y))
   (-
    (* z (+ (* -0.5 (* y y)) (* (pow y 3.0) -0.3333333333333333)))
    (* y (+ z -1.0))))
  t))

C

double code(double x, double y, double z, double t) {
	return (((x + -1.0) * log(y)) + ((z * ((-0.5 * (y * y)) + (pow(y, 3.0) * -0.3333333333333333))) - (y * (z + -1.0)))) - 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 = (((x + (-1.0d0)) * log(y)) + ((z * (((-0.5d0) * (y * y)) + ((y ** 3.0d0) * (-0.3333333333333333d0)))) - (y * (z + (-1.0d0))))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((x + -1.0) * Math.log(y)) + ((z * ((-0.5 * (y * y)) + (Math.pow(y, 3.0) * -0.3333333333333333))) - (y * (z + -1.0)))) - t;
}

Python

def code(x, y, z, t):
	return (((x + -1.0) * math.log(y)) + ((z * ((-0.5 * (y * y)) + (math.pow(y, 3.0) * -0.3333333333333333))) - (y * (z + -1.0)))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x + -1.0) * log(y)) + Float64(Float64(z * Float64(Float64(-0.5 * Float64(y * y)) + Float64((y ^ 3.0) * -0.3333333333333333))) - Float64(y * Float64(z + -1.0)))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x + -1.0) * log(y)) + ((z * ((-0.5 * (y * y)) + ((y ^ 3.0) * -0.3333333333333333))) - (y * (z + -1.0)))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x + -1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[(z * N[(N[(-0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(N[Power[y, 3.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 89.7%

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

2. Taylor expanded in y around 0 99.3%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + \left(-0.3333333333333333 \cdot \left(\left(z - 1\right) \cdot {y}^{3}\right) + -1 \cdot \left(\left(z - 1\right) \cdot y\right)\right)\right)}\right) - t \]
3. Step-by-step derivation

   1. associate-+r+ 99.3%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\left(\left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + -0.3333333333333333 \cdot \left(\left(z - 1\right) \cdot {y}^{3}\right)\right) + -1 \cdot \left(\left(z - 1\right) \cdot y\right)\right)}\right) - t \]

   2. mul-1-neg 99.3%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(\left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + -0.3333333333333333 \cdot \left(\left(z - 1\right) \cdot {y}^{3}\right)\right) + \color{blue}{\left(-\left(z - 1\right) \cdot y\right)}\right)\right) - t \]

   3. unsub-neg 99.3%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\left(\left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + -0.3333333333333333 \cdot \left(\left(z - 1\right) \cdot {y}^{3}\right)\right) - \left(z - 1\right) \cdot y\right)}\right) - t \]

4. Simplified 99.3%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\left(\mathsf{fma}\left(0.3333333333333333 + -0.3333333333333333 \cdot z, {y}^{3}, \left(y \cdot y\right) \cdot \left(0.5 + -0.5 \cdot z\right)\right) - y \cdot \left(-1 + z\right)\right)}\right) - t \]

5. Taylor expanded in z around inf 99.1%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(\color{blue}{\left(-0.5 \cdot {y}^{2} + -0.3333333333333333 \cdot {y}^{3}\right) \cdot z} - y \cdot \left(-1 + z\right)\right)\right) - t \]
6. Step-by-step derivation

   1. *-commutative 99.1%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(\color{blue}{z \cdot \left(-0.5 \cdot {y}^{2} + -0.3333333333333333 \cdot {y}^{3}\right)} - y \cdot \left(-1 + z\right)\right)\right) - t \]

   2. fma-def 99.1%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z \cdot \color{blue}{\mathsf{fma}\left(-0.5, {y}^{2}, -0.3333333333333333 \cdot {y}^{3}\right)} - y \cdot \left(-1 + z\right)\right)\right) - t \]

   3. unpow2 99.1%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z \cdot \mathsf{fma}\left(-0.5, \color{blue}{y \cdot y}, -0.3333333333333333 \cdot {y}^{3}\right) - y \cdot \left(-1 + z\right)\right)\right) - t \]

   4. *-commutative 99.1%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z \cdot \mathsf{fma}\left(-0.5, y \cdot y, \color{blue}{{y}^{3} \cdot -0.3333333333333333}\right) - y \cdot \left(-1 + z\right)\right)\right) - t \]

7. Simplified 99.1%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(\color{blue}{z \cdot \mathsf{fma}\left(-0.5, y \cdot y, {y}^{3} \cdot -0.3333333333333333\right)} - y \cdot \left(-1 + z\right)\right)\right) - t \]
8. Step-by-step derivation

   1. fma-udef 99.1%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z \cdot \color{blue}{\left(-0.5 \cdot \left(y \cdot y\right) + {y}^{3} \cdot -0.3333333333333333\right)} - y \cdot \left(-1 + z\right)\right)\right) - t \]

9. Applied egg-rr 99.1%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z \cdot \color{blue}{\left(-0.5 \cdot \left(y \cdot y\right) + {y}^{3} \cdot -0.3333333333333333\right)} - y \cdot \left(-1 + z\right)\right)\right) - t \]

10. Final simplification 99.1%

    \[\leadsto \left(\left(x + -1\right) \cdot \log y + \left(z \cdot \left(-0.5 \cdot \left(y \cdot y\right) + {y}^{3} \cdot -0.3333333333333333\right) - y \cdot \left(z + -1\right)\right)\right) - t \]

Alternative 3: 99.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (((x + -1.0) * log(y)) + (((y * y) * (0.5 + (z * -0.5))) - (y * (z + -1.0)))) - 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 = (((x + (-1.0d0)) * log(y)) + (((y * y) * (0.5d0 + (z * (-0.5d0)))) - (y * (z + (-1.0d0))))) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

2. Taylor expanded in y around 0 99.1%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + -1 \cdot \left(\left(z - 1\right) \cdot y\right)\right)}\right) - t \]
3. Step-by-step derivation

   1. mul-1-neg 99.1%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + \color{blue}{\left(-\left(z - 1\right) \cdot y\right)}\right)\right) - t \]

   2. unsub-neg 99.1%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) - \left(z - 1\right) \cdot y\right)}\right) - t \]

   3. associate-*r* 99.1%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(\color{blue}{\left(-0.5 \cdot \left(z - 1\right)\right) \cdot {y}^{2}} - \left(z - 1\right) \cdot y\right)\right) - t \]

   4. *-commutative 99.1%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(\color{blue}{{y}^{2} \cdot \left(-0.5 \cdot \left(z - 1\right)\right)} - \left(z - 1\right) \cdot y\right)\right) - t \]

   5. unpow2 99.1%

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

   6. sub-neg 99.1%

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

   7. metadata-eval 99.1%

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

   8. +-commutative 99.1%

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

   9. distribute-lft-in 99.1%

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

   10. metadata-eval 99.1%

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

   11. *-commutative 99.1%

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

   12. sub-neg 99.1%

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

   13. metadata-eval 99.1%

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

   14. +-commutative 99.1%

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

4. Simplified 99.1%

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

5. Final simplification 99.1%

   \[\leadsto \left(\left(x + -1\right) \cdot \log y + \left(\left(y \cdot y\right) \cdot \left(0.5 + z \cdot -0.5\right) - y \cdot \left(z + -1\right)\right)\right) - t \]

Alternative 4: 97.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + -1 \leq -20000000000000 \lor \neg \left(x + -1 \leq -1\right):\\ \;\;\;\;\left(x \cdot \log y - y \cdot \left(z + -1\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y - z \cdot y\right) - \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (+ x -1.0) -20000000000000.0) (not (<= (+ x -1.0) -1.0)))
   (- (- (* x (log y)) (* y (+ z -1.0))) t)
   (- (- (- y (* z y)) (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x + -1.0) <= -20000000000000.0) || !((x + -1.0) <= -1.0)) {
		tmp = ((x * log(y)) - (y * (z + -1.0))) - t;
	} else {
		tmp = ((y - (z * y)) - log(y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x + (-1.0d0)) <= (-20000000000000.0d0)) .or. (.not. ((x + (-1.0d0)) <= (-1.0d0)))) then
        tmp = ((x * log(y)) - (y * (z + (-1.0d0)))) - t
    else
        tmp = ((y - (z * y)) - log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x + -1.0) <= -20000000000000.0) || !((x + -1.0) <= -1.0)) {
		tmp = ((x * Math.log(y)) - (y * (z + -1.0))) - t;
	} else {
		tmp = ((y - (z * y)) - Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x + -1.0) <= -20000000000000.0) or not ((x + -1.0) <= -1.0):
		tmp = ((x * math.log(y)) - (y * (z + -1.0))) - t
	else:
		tmp = ((y - (z * y)) - math.log(y)) - t
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x + -1.0), $MachinePrecision], -20000000000000.0], N[Not[LessEqual[N[(x + -1.0), $MachinePrecision], -1.0]], $MachinePrecision]], N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - N[(y * N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(y - N[(z * y), $MachinePrecision]), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x 1) < -2e13 or -1 < (-.f64 x 1)

   1. Initial program 92.5%

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

   2. Taylor expanded in y around 0 98.3%

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

      1. +-commutative 98.3%

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

      2. sub-neg 98.3%

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

      3. metadata-eval 98.3%

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

      4. mul-1-neg 98.3%

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

      5. unsub-neg 98.3%

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

      6. *-commutative 98.3%

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

      7. +-commutative 98.3%

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

      8. *-commutative 98.3%

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

      9. sub-neg 98.3%

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

      10. metadata-eval 98.3%

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

      11. +-commutative 98.3%

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

   4. Simplified 98.3%

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

   5. Taylor expanded in x around inf 98.2%

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

   if -2e13 < (-.f64 x 1) < -1

   1. Initial program 87.0%

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

      1. add-cbrt-cube 86.7%

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

      2. pow3 86.8%

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

      3. sub-neg 86.8%

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

      4. metadata-eval 86.8%

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

   3. Applied egg-rr 86.8%

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

   4. Taylor expanded in y around 0 98.3%

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

      1. mul-1-neg 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in x around 0 98.3%

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

      1. mul-1-neg 98.3%

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

      2. sub-neg 98.3%

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

      3. metadata-eval 98.3%

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

      4. *-commutative 98.3%

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

      5. distribute-lft-neg-out 98.3%

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

      6. mul-1-neg 98.3%

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

      7. unsub-neg 98.3%

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

      8. +-commutative 98.3%

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

      9. distribute-rgt-in 98.3%

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

      10. neg-mul-1 98.3%

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

      11. remove-double-neg 98.3%

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

      12. distribute-rgt-neg-in 98.3%

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

      13. unsub-neg 98.3%

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

      14. *-commutative 98.3%

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

   9. Simplified 98.3%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + -1 \leq -20000000000000 \lor \neg \left(x + -1 \leq -1\right):\\ \;\;\;\;\left(x \cdot \log y - y \cdot \left(z + -1\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y - z \cdot y\right) - \log y\right) - t\\ \end{array} \]

Alternative 5: 99.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (((x + -1.0) * log(y)) + ((y * (y * 0.5)) - (y * (z + -1.0)))) - 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 = (((x + (-1.0d0)) * log(y)) + ((y * (y * 0.5d0)) - (y * (z + (-1.0d0))))) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

2. Taylor expanded in y around 0 99.1%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + -1 \cdot \left(\left(z - 1\right) \cdot y\right)\right)}\right) - t \]
3. Step-by-step derivation

   1. mul-1-neg 99.1%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + \color{blue}{\left(-\left(z - 1\right) \cdot y\right)}\right)\right) - t \]

   2. unsub-neg 99.1%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) - \left(z - 1\right) \cdot y\right)}\right) - t \]

   3. associate-*r* 99.1%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(\color{blue}{\left(-0.5 \cdot \left(z - 1\right)\right) \cdot {y}^{2}} - \left(z - 1\right) \cdot y\right)\right) - t \]

   4. *-commutative 99.1%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(\color{blue}{{y}^{2} \cdot \left(-0.5 \cdot \left(z - 1\right)\right)} - \left(z - 1\right) \cdot y\right)\right) - t \]

   5. unpow2 99.1%

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

   6. sub-neg 99.1%

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

   7. metadata-eval 99.1%

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

   8. +-commutative 99.1%

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

   9. distribute-lft-in 99.1%

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

   10. metadata-eval 99.1%

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

   11. *-commutative 99.1%

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

   12. sub-neg 99.1%

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

   13. metadata-eval 99.1%

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

   14. +-commutative 99.1%

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

4. Simplified 99.1%

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

5. Taylor expanded in z around 0 98.5%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(\color{blue}{0.5 \cdot {y}^{2}} - y \cdot \left(-1 + z\right)\right)\right) - t \]
6. Step-by-step derivation

   1. unpow2 98.5%

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

   2. *-commutative 98.5%

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

   3. associate-*l* 98.5%

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

7. Simplified 98.5%

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

8. Final simplification 98.5%

   \[\leadsto \left(\left(x + -1\right) \cdot \log y + \left(y \cdot \left(y \cdot 0.5\right) - y \cdot \left(z + -1\right)\right)\right) - t \]

Alternative 6: 95.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + -1 \leq -2 \cdot 10^{+31}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x + -1 \leq -1:\\ \;\;\;\;\left(\left(y - z \cdot y\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(x + -1\right) \cdot \log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x -1.0) -2e+31)
   (- (* x (log y)) t)
   (if (<= (+ x -1.0) -1.0)
     (- (- (- y (* z y)) (log y)) t)
     (- (* (+ x -1.0) (log y)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + -1.0) <= -2e+31) {
		tmp = (x * log(y)) - t;
	} else if ((x + -1.0) <= -1.0) {
		tmp = ((y - (z * y)) - log(y)) - t;
	} else {
		tmp = ((x + -1.0) * log(y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x + (-1.0d0)) <= (-2d+31)) then
        tmp = (x * log(y)) - t
    else if ((x + (-1.0d0)) <= (-1.0d0)) then
        tmp = ((y - (z * y)) - log(y)) - t
    else
        tmp = ((x + (-1.0d0)) * log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + -1.0) <= -2e+31) {
		tmp = (x * Math.log(y)) - t;
	} else if ((x + -1.0) <= -1.0) {
		tmp = ((y - (z * y)) - Math.log(y)) - t;
	} else {
		tmp = ((x + -1.0) * Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + -1.0) <= -2e+31:
		tmp = (x * math.log(y)) - t
	elif (x + -1.0) <= -1.0:
		tmp = ((y - (z * y)) - math.log(y)) - t
	else:
		tmp = ((x + -1.0) * math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + -1.0) <= -2e+31)
		tmp = Float64(Float64(x * log(y)) - t);
	elseif (Float64(x + -1.0) <= -1.0)
		tmp = Float64(Float64(Float64(y - Float64(z * y)) - log(y)) - t);
	else
		tmp = Float64(Float64(Float64(x + -1.0) * log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + -1.0) <= -2e+31)
		tmp = (x * log(y)) - t;
	elseif ((x + -1.0) <= -1.0)
		tmp = ((y - (z * y)) - log(y)) - t;
	else
		tmp = ((x + -1.0) * log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + -1.0), $MachinePrecision], -2e+31], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(x + -1.0), $MachinePrecision], -1.0], N[(N[(N[(y - N[(z * y), $MachinePrecision]), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(x + -1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x 1) < -1.9999999999999999e31

   1. Initial program 93.6%

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

   2. Taylor expanded in y around 0 98.9%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + -1 \cdot \left(\left(z - 1\right) \cdot y\right)\right)}\right) - t \]
   3. Step-by-step derivation

      1. mul-1-neg 98.9%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + \color{blue}{\left(-\left(z - 1\right) \cdot y\right)}\right)\right) - t \]

      2. unsub-neg 98.9%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) - \left(z - 1\right) \cdot y\right)}\right) - t \]

      3. associate-*r* 98.9%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(\color{blue}{\left(-0.5 \cdot \left(z - 1\right)\right) \cdot {y}^{2}} - \left(z - 1\right) \cdot y\right)\right) - t \]

      4. *-commutative 98.9%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(\color{blue}{{y}^{2} \cdot \left(-0.5 \cdot \left(z - 1\right)\right)} - \left(z - 1\right) \cdot y\right)\right) - t \]

      5. unpow2 98.9%

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

      6. sub-neg 98.9%

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

      7. metadata-eval 98.9%

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

      8. +-commutative 98.9%

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

      9. distribute-lft-in 98.9%

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

      10. metadata-eval 98.9%

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

      11. *-commutative 98.9%

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

      12. sub-neg 98.9%

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

      13. metadata-eval 98.9%

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

      14. +-commutative 98.9%

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

   4. Simplified 98.9%

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

      1. *-commutative 98.9%

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

      2. flip-- 51.7%

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

      3. associate-*r/ 48.8%

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

      4. metadata-eval 48.8%

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

      5. fma-neg 48.8%

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

      6. metadata-eval 48.8%

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

      7. +-commutative 48.8%

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

   6. Applied egg-rr 48.8%

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

      1. associate-/l* 51.6%

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

   8. Simplified 51.6%

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

   9. Taylor expanded in x around inf 98.8%

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

   10. Taylor expanded in y around 0 90.9%

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

   if -1.9999999999999999e31 < (-.f64 x 1) < -1

   1. Initial program 85.7%

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

      1. add-cbrt-cube 85.4%

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

      2. pow3 85.5%

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

      3. sub-neg 85.5%

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

      4. metadata-eval 85.5%

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

   3. Applied egg-rr 85.5%

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

   4. Taylor expanded in y around 0 98.3%

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

      1. mul-1-neg 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in x around 0 97.8%

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

      1. mul-1-neg 97.8%

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

      2. sub-neg 97.8%

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

      3. metadata-eval 97.8%

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

      4. *-commutative 97.8%

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

      5. distribute-lft-neg-out 97.8%

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

      6. mul-1-neg 97.8%

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

      7. unsub-neg 97.8%

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

      8. +-commutative 97.8%

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

      9. distribute-rgt-in 97.8%

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

      10. neg-mul-1 97.8%

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

      11. remove-double-neg 97.8%

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

      12. distribute-rgt-neg-in 97.8%

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

      13. unsub-neg 97.8%

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

      14. *-commutative 97.8%

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

   9. Simplified 97.8%

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

   if -1 < (-.f64 x 1)

   1. Initial program 94.5%

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

   2. Taylor expanded in y around 0 93.0%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + -1 \leq -2 \cdot 10^{+31}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x + -1 \leq -1:\\ \;\;\;\;\left(\left(y - z \cdot y\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(x + -1\right) \cdot \log y - t\\ \end{array} \]

Alternative 7: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

2. Taylor expanded in y around 0 98.4%

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

   1. +-commutative 98.4%

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

   2. sub-neg 98.4%

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

   3. metadata-eval 98.4%

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

   4. mul-1-neg 98.4%

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

   5. unsub-neg 98.4%

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

   6. *-commutative 98.4%

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

   7. +-commutative 98.4%

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

   8. *-commutative 98.4%

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

   9. sub-neg 98.4%

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

   10. metadata-eval 98.4%

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

   11. +-commutative 98.4%

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

4. Simplified 98.4%

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

5. Final simplification 98.4%

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

Alternative 8: 76.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+24} \lor \neg \left(x \leq 4.2 \cdot 10^{-13}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -4.5e+24) (not (<= x 4.2e-13)))
   (- (* x (log y)) t)
   (- (* z (log1p (- y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.5e+24) || !(x <= 4.2e-13)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (z * log1p(-y)) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.5e+24) || !(x <= 4.2e-13)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (z * Math.log1p(-y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -4.5e+24) or not (x <= 4.2e-13):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (z * math.log1p(-y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -4.5e+24) || !(x <= 4.2e-13))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -4.5e+24], N[Not[LessEqual[x, 4.2e-13]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.50000000000000019e24 or 4.19999999999999977e-13 < x

   1. Initial program 94.0%

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

   2. Taylor expanded in y around 0 98.9%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + -1 \cdot \left(\left(z - 1\right) \cdot y\right)\right)}\right) - t \]
   3. Step-by-step derivation

      1. mul-1-neg 98.9%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + \color{blue}{\left(-\left(z - 1\right) \cdot y\right)}\right)\right) - t \]

      2. unsub-neg 98.9%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) - \left(z - 1\right) \cdot y\right)}\right) - t \]

      3. associate-*r* 98.9%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(\color{blue}{\left(-0.5 \cdot \left(z - 1\right)\right) \cdot {y}^{2}} - \left(z - 1\right) \cdot y\right)\right) - t \]

      4. *-commutative 98.9%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(\color{blue}{{y}^{2} \cdot \left(-0.5 \cdot \left(z - 1\right)\right)} - \left(z - 1\right) \cdot y\right)\right) - t \]

      5. unpow2 98.9%

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

      6. sub-neg 98.9%

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

      7. metadata-eval 98.9%

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

      8. +-commutative 98.9%

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

      9. distribute-lft-in 98.9%

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

      10. metadata-eval 98.9%

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

      11. *-commutative 98.9%

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

      12. sub-neg 98.9%

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

      13. metadata-eval 98.9%

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

      14. +-commutative 98.9%

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

   4. Simplified 98.9%

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

      1. *-commutative 98.9%

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

      2. flip-- 55.1%

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

      3. associate-*r/ 53.5%

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

      4. metadata-eval 53.5%

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

      5. fma-neg 53.5%

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

      6. metadata-eval 53.5%

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

      7. +-commutative 53.5%

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

   6. Applied egg-rr 53.5%

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

      1. associate-/l* 55.0%

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

   8. Simplified 55.0%

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

   9. Taylor expanded in x around inf 98.8%

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

   10. Taylor expanded in y around 0 91.8%

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

   if -4.50000000000000019e24 < x < 4.19999999999999977e-13

   1. Initial program 85.7%

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

   2. Taylor expanded in z around inf 49.8%

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

      1. sub-neg 49.8%

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

      2. mul-1-neg 49.8%

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

      3. log1p-def 62.9%

         \[\leadsto z \cdot \color{blue}{\mathsf{log1p}\left(-1 \cdot y\right)} - t \]

      4. mul-1-neg 62.9%

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

   4. Simplified 62.9%

      \[\leadsto \color{blue}{z \cdot \mathsf{log1p}\left(-y\right)} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+24} \lor \neg \left(x \leq 4.2 \cdot 10^{-13}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]

Alternative 9: 76.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+24} \lor \neg \left(x \leq 4.2 \cdot 10^{-13}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-0.5 \cdot \left(y \cdot y\right) - y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.7e+24) (not (<= x 4.2e-13)))
   (- (* x (log y)) t)
   (- (* z (- (* -0.5 (* y y)) y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.7e+24) || !(x <= 4.2e-13)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (z * ((-0.5 * (y * y)) - y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-2.7d+24)) .or. (.not. (x <= 4.2d-13))) then
        tmp = (x * log(y)) - t
    else
        tmp = (z * (((-0.5d0) * (y * y)) - y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.7e+24) || !(x <= 4.2e-13)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (z * ((-0.5 * (y * y)) - y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2.7e+24) or not (x <= 4.2e-13):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (z * ((-0.5 * (y * y)) - y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.7e+24) || !(x <= 4.2e-13))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(z * Float64(Float64(-0.5 * Float64(y * y)) - y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -2.7e+24) || ~((x <= 4.2e-13)))
		tmp = (x * log(y)) - t;
	else
		tmp = (z * ((-0.5 * (y * y)) - y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.7e+24], N[Not[LessEqual[x, 4.2e-13]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(z * N[(N[(-0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.7e24 or 4.19999999999999977e-13 < x

   1. Initial program 94.0%

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

   2. Taylor expanded in y around 0 98.9%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + -1 \cdot \left(\left(z - 1\right) \cdot y\right)\right)}\right) - t \]
   3. Step-by-step derivation

      1. mul-1-neg 98.9%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + \color{blue}{\left(-\left(z - 1\right) \cdot y\right)}\right)\right) - t \]

      2. unsub-neg 98.9%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) - \left(z - 1\right) \cdot y\right)}\right) - t \]

      3. associate-*r* 98.9%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(\color{blue}{\left(-0.5 \cdot \left(z - 1\right)\right) \cdot {y}^{2}} - \left(z - 1\right) \cdot y\right)\right) - t \]

      4. *-commutative 98.9%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(\color{blue}{{y}^{2} \cdot \left(-0.5 \cdot \left(z - 1\right)\right)} - \left(z - 1\right) \cdot y\right)\right) - t \]

      5. unpow2 98.9%

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

      6. sub-neg 98.9%

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

      7. metadata-eval 98.9%

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

      8. +-commutative 98.9%

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

      9. distribute-lft-in 98.9%

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

      10. metadata-eval 98.9%

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

      11. *-commutative 98.9%

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

      12. sub-neg 98.9%

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

      13. metadata-eval 98.9%

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

      14. +-commutative 98.9%

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

   4. Simplified 98.9%

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

      1. *-commutative 98.9%

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

      2. flip-- 55.1%

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

      3. associate-*r/ 53.5%

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

      4. metadata-eval 53.5%

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

      5. fma-neg 53.5%

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

      6. metadata-eval 53.5%

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

      7. +-commutative 53.5%

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

   6. Applied egg-rr 53.5%

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

      1. associate-/l* 55.0%

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

   8. Simplified 55.0%

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

   9. Taylor expanded in x around inf 98.8%

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

   10. Taylor expanded in y around 0 91.8%

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

   if -2.7e24 < x < 4.19999999999999977e-13

   1. Initial program 85.7%

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

   2. Taylor expanded in y around 0 99.2%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + -1 \cdot \left(\left(z - 1\right) \cdot y\right)\right)}\right) - t \]
   3. Step-by-step derivation

      1. mul-1-neg 99.2%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + \color{blue}{\left(-\left(z - 1\right) \cdot y\right)}\right)\right) - t \]

      2. unsub-neg 99.2%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) - \left(z - 1\right) \cdot y\right)}\right) - t \]

      3. associate-*r* 99.2%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(\color{blue}{\left(-0.5 \cdot \left(z - 1\right)\right) \cdot {y}^{2}} - \left(z - 1\right) \cdot y\right)\right) - t \]

      4. *-commutative 99.2%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(\color{blue}{{y}^{2} \cdot \left(-0.5 \cdot \left(z - 1\right)\right)} - \left(z - 1\right) \cdot y\right)\right) - t \]

      5. unpow2 99.2%

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

      6. sub-neg 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

      9. distribute-lft-in 99.2%

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

      10. metadata-eval 99.2%

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

      11. *-commutative 99.2%

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

      12. sub-neg 99.2%

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

      13. metadata-eval 99.2%

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

      14. +-commutative 99.2%

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

   4. Simplified 99.2%

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

      1. *-commutative 99.2%

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

      2. flip-- 99.2%

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

      3. associate-*r/ 99.2%

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

      4. metadata-eval 99.2%

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

      5. fma-neg 99.2%

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

      6. metadata-eval 99.2%

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

      7. +-commutative 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. associate-/l* 99.2%

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

   8. Simplified 99.2%

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

   9. Taylor expanded in z around inf 62.8%

      \[\leadsto \color{blue}{\left(-0.5 \cdot {y}^{2} - y\right) \cdot z} - t \]
   10. Step-by-step derivation

       1. *-commutative 62.8%

          \[\leadsto \color{blue}{z \cdot \left(-0.5 \cdot {y}^{2} - y\right)} - t \]

       2. unpow2 62.8%

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

   11. Simplified 62.8%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+24} \lor \neg \left(x \leq 4.2 \cdot 10^{-13}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-0.5 \cdot \left(y \cdot y\right) - y\right) - t\\ \end{array} \]

Alternative 10: 88.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+275}:\\ \;\;\;\;z \cdot \left(-0.5 \cdot \left(y \cdot y\right) - y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(x + -1\right) \cdot \log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2e+275)
   (- (* z (- (* -0.5 (* y y)) y)) t)
   (- (* (+ x -1.0) (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2e+275) {
		tmp = (z * ((-0.5 * (y * y)) - y)) - t;
	} else {
		tmp = ((x + -1.0) * log(y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-2d+275)) then
        tmp = (z * (((-0.5d0) * (y * y)) - y)) - t
    else
        tmp = ((x + (-1.0d0)) * log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2e+275) {
		tmp = (z * ((-0.5 * (y * y)) - y)) - t;
	} else {
		tmp = ((x + -1.0) * Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -2e+275:
		tmp = (z * ((-0.5 * (y * y)) - y)) - t
	else:
		tmp = ((x + -1.0) * math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2e+275)
		tmp = Float64(Float64(z * Float64(Float64(-0.5 * Float64(y * y)) - y)) - t);
	else
		tmp = Float64(Float64(Float64(x + -1.0) * log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -2e+275)
		tmp = (z * ((-0.5 * (y * y)) - y)) - t;
	else
		tmp = ((x + -1.0) * log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2e+275], N[(N[(z * N[(N[(-0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(x + -1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.99999999999999992e275

   1. Initial program 22.9%

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

   2. Taylor expanded in y around 0 99.6%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + -1 \cdot \left(\left(z - 1\right) \cdot y\right)\right)}\right) - t \]
   3. Step-by-step derivation

      1. mul-1-neg 99.6%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + \color{blue}{\left(-\left(z - 1\right) \cdot y\right)}\right)\right) - t \]

      2. unsub-neg 99.6%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) - \left(z - 1\right) \cdot y\right)}\right) - t \]

      3. associate-*r* 99.6%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(\color{blue}{\left(-0.5 \cdot \left(z - 1\right)\right) \cdot {y}^{2}} - \left(z - 1\right) \cdot y\right)\right) - t \]

      4. *-commutative 99.6%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(\color{blue}{{y}^{2} \cdot \left(-0.5 \cdot \left(z - 1\right)\right)} - \left(z - 1\right) \cdot y\right)\right) - t \]

      5. unpow2 99.6%

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

      6. sub-neg 99.6%

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

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. distribute-lft-in 99.6%

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

      10. metadata-eval 99.6%

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

      11. *-commutative 99.6%

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

      12. sub-neg 99.6%

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

      13. metadata-eval 99.6%

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

      14. +-commutative 99.6%

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

   4. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. flip-- 99.6%

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

      3. associate-*r/ 99.6%

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

      4. metadata-eval 99.6%

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

      5. fma-neg 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-/l* 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in z around inf 99.8%

      \[\leadsto \color{blue}{\left(-0.5 \cdot {y}^{2} - y\right) \cdot z} - t \]
   10. Step-by-step derivation

       1. *-commutative 99.8%

          \[\leadsto \color{blue}{z \cdot \left(-0.5 \cdot {y}^{2} - y\right)} - t \]

       2. unpow2 99.8%

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

   11. Simplified 99.8%

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

   if -1.99999999999999992e275 < z

   1. Initial program 91.8%

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

   2. Taylor expanded in y around 0 89.8%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+275}:\\ \;\;\;\;z \cdot \left(-0.5 \cdot \left(y \cdot y\right) - y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(x + -1\right) \cdot \log y - t\\ \end{array} \]

Alternative 11: 45.2% accurate, 19.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y, z, t):
	return (z * ((-0.5 * (y * y)) - y)) - t

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = (z * ((-0.5 * (y * y)) - y)) - t;
end

Wolfram

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

Derivation

1. Initial program 89.7%

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

2. Taylor expanded in y around 0 99.1%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + -1 \cdot \left(\left(z - 1\right) \cdot y\right)\right)}\right) - t \]
3. Step-by-step derivation

   1. mul-1-neg 99.1%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + \color{blue}{\left(-\left(z - 1\right) \cdot y\right)}\right)\right) - t \]

   2. unsub-neg 99.1%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) - \left(z - 1\right) \cdot y\right)}\right) - t \]

   3. associate-*r* 99.1%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(\color{blue}{\left(-0.5 \cdot \left(z - 1\right)\right) \cdot {y}^{2}} - \left(z - 1\right) \cdot y\right)\right) - t \]

   4. *-commutative 99.1%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(\color{blue}{{y}^{2} \cdot \left(-0.5 \cdot \left(z - 1\right)\right)} - \left(z - 1\right) \cdot y\right)\right) - t \]

   5. unpow2 99.1%

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

   6. sub-neg 99.1%

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

   7. metadata-eval 99.1%

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

   8. +-commutative 99.1%

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

   9. distribute-lft-in 99.1%

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

   10. metadata-eval 99.1%

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

   11. *-commutative 99.1%

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

   12. sub-neg 99.1%

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

   13. metadata-eval 99.1%

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

   14. +-commutative 99.1%

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

4. Simplified 99.1%

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

   1. *-commutative 99.1%

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

   2. flip-- 78.0%

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

   3. associate-*r/ 77.2%

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

   4. metadata-eval 77.2%

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

   5. fma-neg 77.2%

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

   6. metadata-eval 77.2%

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

   7. +-commutative 77.2%

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

6. Applied egg-rr 77.2%

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

   1. associate-/l* 78.0%

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

8. Simplified 78.0%

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

9. Taylor expanded in z around inf 47.0%

   \[\leadsto \color{blue}{\left(-0.5 \cdot {y}^{2} - y\right) \cdot z} - t \]
10. Step-by-step derivation

    1. *-commutative 47.0%

       \[\leadsto \color{blue}{z \cdot \left(-0.5 \cdot {y}^{2} - y\right)} - t \]

    2. unpow2 47.0%

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

11. Simplified 47.0%

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

12. Final simplification 47.0%

    \[\leadsto z \cdot \left(-0.5 \cdot \left(y \cdot y\right) - y\right) - t \]

Alternative 12: 45.2% accurate, 30.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y, z, t):
	return (y * (1.0 - z)) - t

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

2. Taylor expanded in y around 0 98.4%

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

   1. +-commutative 98.4%

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

   2. sub-neg 98.4%

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

   3. metadata-eval 98.4%

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

   4. mul-1-neg 98.4%

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

   5. unsub-neg 98.4%

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

   6. *-commutative 98.4%

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

   7. +-commutative 98.4%

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

   8. *-commutative 98.4%

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

   9. sub-neg 98.4%

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

   10. metadata-eval 98.4%

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

   11. +-commutative 98.4%

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

4. Simplified 98.4%

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

5. Taylor expanded in y around inf 46.7%

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

6. Final simplification 46.7%

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

Alternative 13: 45.0% accurate, 35.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

2. Taylor expanded in y around 0 98.4%

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

   1. +-commutative 98.4%

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

   2. sub-neg 98.4%

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

   3. metadata-eval 98.4%

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

   4. mul-1-neg 98.4%

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

   5. unsub-neg 98.4%

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

   6. *-commutative 98.4%

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

   7. +-commutative 98.4%

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

   8. *-commutative 98.4%

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

   9. sub-neg 98.4%

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

   10. metadata-eval 98.4%

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

   11. +-commutative 98.4%

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

4. Simplified 98.4%

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

5. Taylor expanded in z around inf 46.5%

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

   1. associate-*r* 46.5%

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

   2. neg-mul-1 46.5%

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

7. Simplified 46.5%

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

8. Final simplification 46.5%

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

Alternative 14: 35.1% accurate, 107.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := (-t)

Derivation

1. Initial program 89.7%

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

   1. +-commutative 89.7%

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

   2. associate--l+ 89.7%

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

   3. fma-def 89.7%

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

   4. sub-neg 89.7%

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

   5. log1p-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around inf 36.3%

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

   1. neg-mul-1 36.3%

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

6. Simplified 36.3%

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

7. Final simplification 36.3%

   \[\leadsto -t \]

Reproduce

herbie shell --seed 2023257 
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  :precision binary64
  (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))