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

Percentage Accurate: 89.5% → 99.8%

Time: 24.5s

Alternatives: 15

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: 89.5% 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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

   1. +-commutative 88.0%

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

   2. associate--l+ 88.0%

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

   3. fma-define 88.0%

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

   4. sub-neg 88.0%

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

   5. metadata-eval 88.0%

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

   6. sub-neg 88.0%

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

   7. log1p-define 99.9%

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

   8. fma-neg 99.9%

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

   9. sub-neg 99.9%

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

   10. metadata-eval 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z + -1, \mathsf{log1p}\left(-y\right), \mathsf{fma}\left(x + -1, \log y, -t\right)\right)} \]
4. Add Preprocessing

5. Final simplification 99.9%

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

Alternative 2: 98.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z + -1\right) \cdot y\\ \mathbf{if}\;-1 + x \leq -20 \lor \neg \left(-1 + x \leq -0.9998\right):\\ \;\;\;\;\left(x \cdot \log y - t\_1\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \left(\log y + t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (+ z -1.0) y)))
   (if (or (<= (+ -1.0 x) -20.0) (not (<= (+ -1.0 x) -0.9998)))
     (- (- (* x (log y)) t_1) t)
     (- (- t) (+ (log y) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z + -1.0) * y;
	double tmp;
	if (((-1.0 + x) <= -20.0) || !((-1.0 + x) <= -0.9998)) {
		tmp = ((x * log(y)) - t_1) - t;
	} else {
		tmp = -t - (log(y) + t_1);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z + (-1.0d0)) * y
    if ((((-1.0d0) + x) <= (-20.0d0)) .or. (.not. (((-1.0d0) + x) <= (-0.9998d0)))) then
        tmp = ((x * log(y)) - t_1) - t
    else
        tmp = -t - (log(y) + t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z + -1.0) * y;
	double tmp;
	if (((-1.0 + x) <= -20.0) || !((-1.0 + x) <= -0.9998)) {
		tmp = ((x * Math.log(y)) - t_1) - t;
	} else {
		tmp = -t - (Math.log(y) + t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z + -1.0) * y
	tmp = 0
	if ((-1.0 + x) <= -20.0) or not ((-1.0 + x) <= -0.9998):
		tmp = ((x * math.log(y)) - t_1) - t
	else:
		tmp = -t - (math.log(y) + t_1)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z + -1.0) * y)
	tmp = 0.0
	if ((Float64(-1.0 + x) <= -20.0) || !(Float64(-1.0 + x) <= -0.9998))
		tmp = Float64(Float64(Float64(x * log(y)) - t_1) - t);
	else
		tmp = Float64(Float64(-t) - Float64(log(y) + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z + -1.0) * y;
	tmp = 0.0;
	if (((-1.0 + x) <= -20.0) || ~(((-1.0 + x) <= -0.9998)))
		tmp = ((x * log(y)) - t_1) - t;
	else
		tmp = -t - (log(y) + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z + -1.0), $MachinePrecision] * y), $MachinePrecision]}, If[Or[LessEqual[N[(-1.0 + x), $MachinePrecision], -20.0], N[Not[LessEqual[N[(-1.0 + x), $MachinePrecision], -0.9998]], $MachinePrecision]], N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] - t), $MachinePrecision], N[((-t) - N[(N[Log[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x 1) < -20 or -0.99980000000000002 < (-.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. Add Preprocessing

   3. Taylor expanded in y around 0 99.8%

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

      1. mul-1-neg 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around inf 98.8%

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

      1. *-commutative 98.8%

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

   8. Simplified 98.8%

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

      1. *-commutative 98.8%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 93.5%

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

      4. sqr-neg 93.5%

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

      5. sqrt-unprod 93.4%

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

      6. add-sqr-sqrt 93.4%

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

      7. cancel-sign-sub 93.4%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 98.0%

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

      10. sqr-neg 98.0%

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

      11. sqrt-unprod 98.7%

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

      12. add-sqr-sqrt 98.8%

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

      13. sub-neg 98.8%

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

      14. metadata-eval 98.8%

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

   10. Applied egg-rr 98.8%

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

   if -20 < (-.f64 x 1) < -0.99980000000000002

   1. Initial program 81.1%

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

   3. Taylor expanded in y around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

      1. flip3-- 100.0%

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

      2. associate-*l/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-define 100.0%

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

      8. metadata-eval 100.0%

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

      9. *-rgt-identity 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. mul-1-neg 99.7%

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

      3. unsub-neg 99.7%

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

      4. mul-1-neg 99.7%

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

      5. sub-neg 99.7%

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

      6. metadata-eval 99.7%

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

      7. *-commutative 99.7%

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

      8. distribute-rgt-neg-in 99.7%

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

      9. +-commutative 99.7%

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

   10. Simplified 99.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -20 \lor \neg \left(-1 + x \leq -0.9998\right):\\ \;\;\;\;\left(x \cdot \log y - \left(z + -1\right) \cdot y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \left(\log y + \left(z + -1\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + x \leq -5000000:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;-1 + x \leq -1:\\ \;\;\;\;\left(-t\right) - \left(\log y + \left(z + -1\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ -1.0 x) -5000000.0)
   (- (* x (log y)) t)
   (if (<= (+ -1.0 x) -1.0)
     (- (- t) (+ (log y) (* (+ z -1.0) y)))
     (- (* (log y) (+ -1.0 x)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((-1.0 + x) <= -5000000.0) {
		tmp = (x * log(y)) - t;
	} else if ((-1.0 + x) <= -1.0) {
		tmp = -t - (log(y) + ((z + -1.0) * y));
	} else {
		tmp = (log(y) * (-1.0 + x)) - 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 (((-1.0d0) + x) <= (-5000000.0d0)) then
        tmp = (x * log(y)) - t
    else if (((-1.0d0) + x) <= (-1.0d0)) then
        tmp = -t - (log(y) + ((z + (-1.0d0)) * y))
    else
        tmp = (log(y) * ((-1.0d0) + x)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((-1.0 + x) <= -5000000.0) {
		tmp = (x * Math.log(y)) - t;
	} else if ((-1.0 + x) <= -1.0) {
		tmp = -t - (Math.log(y) + ((z + -1.0) * y));
	} else {
		tmp = (Math.log(y) * (-1.0 + x)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (-1.0 + x) <= -5000000.0:
		tmp = (x * math.log(y)) - t
	elif (-1.0 + x) <= -1.0:
		tmp = -t - (math.log(y) + ((z + -1.0) * y))
	else:
		tmp = (math.log(y) * (-1.0 + x)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(-1.0 + x) <= -5000000.0)
		tmp = Float64(Float64(x * log(y)) - t);
	elseif (Float64(-1.0 + x) <= -1.0)
		tmp = Float64(Float64(-t) - Float64(log(y) + Float64(Float64(z + -1.0) * y)));
	else
		tmp = Float64(Float64(log(y) * Float64(-1.0 + x)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((-1.0 + x) <= -5000000.0)
		tmp = (x * log(y)) - t;
	elseif ((-1.0 + x) <= -1.0)
		tmp = -t - (log(y) + ((z + -1.0) * y));
	else
		tmp = (log(y) * (-1.0 + x)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(-1.0 + x), $MachinePrecision], -5000000.0], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(-1.0 + x), $MachinePrecision], -1.0], N[((-t) - N[(N[Log[y], $MachinePrecision] + N[(N[(z + -1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x 1) < -5e6

   1. Initial program 97.9%

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

   3. Taylor expanded in y around 0 99.8%

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

      1. mul-1-neg 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around inf 97.9%

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

      1. *-commutative 97.9%

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

   8. Simplified 97.9%

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

   if -5e6 < (-.f64 x 1) < -1

   1. Initial program 80.7%

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

   3. Taylor expanded in y around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

      1. flip3-- 100.0%

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

      2. associate-*l/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-define 100.0%

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

      8. metadata-eval 100.0%

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

      9. *-rgt-identity 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around 0 99.2%

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

      1. +-commutative 99.2%

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

      2. mul-1-neg 99.2%

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

      3. unsub-neg 99.2%

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

      4. mul-1-neg 99.2%

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

      5. sub-neg 99.2%

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

      6. metadata-eval 99.2%

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

      7. *-commutative 99.2%

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

      8. distribute-rgt-neg-in 99.2%

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

      9. +-commutative 99.2%

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

   10. Simplified 99.2%

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

   if -1 < (-.f64 x 1)

   1. Initial program 93.2%

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

      1. +-commutative 93.2%

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

      2. fma-define 93.2%

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

      3. sub-neg 93.2%

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

      4. metadata-eval 93.2%

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

      5. sub-neg 93.2%

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

      6. log1p-define 99.8%

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

      7. sub-neg 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 93.2%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -5000000:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;-1 + x \leq -1:\\ \;\;\;\;\left(-t\right) - \left(\log y + \left(z + -1\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + x \leq -5000000:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;-1 + x \leq -1:\\ \;\;\;\;\left(-t\right) - \left(\log y + \left(z + -1\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \left(-1 + x\right) \cdot \log \left(\frac{1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ -1.0 x) -5000000.0)
   (- (* x (log y)) t)
   (if (<= (+ -1.0 x) -1.0)
     (- (- t) (+ (log y) (* (+ z -1.0) y)))
     (- (- t) (* (+ -1.0 x) (log (/ 1.0 y)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((-1.0 + x) <= -5000000.0) {
		tmp = (x * log(y)) - t;
	} else if ((-1.0 + x) <= -1.0) {
		tmp = -t - (log(y) + ((z + -1.0) * y));
	} else {
		tmp = -t - ((-1.0 + x) * log((1.0 / y)));
	}
	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 (((-1.0d0) + x) <= (-5000000.0d0)) then
        tmp = (x * log(y)) - t
    else if (((-1.0d0) + x) <= (-1.0d0)) then
        tmp = -t - (log(y) + ((z + (-1.0d0)) * y))
    else
        tmp = -t - (((-1.0d0) + x) * log((1.0d0 / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((-1.0 + x) <= -5000000.0) {
		tmp = (x * Math.log(y)) - t;
	} else if ((-1.0 + x) <= -1.0) {
		tmp = -t - (Math.log(y) + ((z + -1.0) * y));
	} else {
		tmp = -t - ((-1.0 + x) * Math.log((1.0 / y)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (-1.0 + x) <= -5000000.0:
		tmp = (x * math.log(y)) - t
	elif (-1.0 + x) <= -1.0:
		tmp = -t - (math.log(y) + ((z + -1.0) * y))
	else:
		tmp = -t - ((-1.0 + x) * math.log((1.0 / y)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(-1.0 + x) <= -5000000.0)
		tmp = Float64(Float64(x * log(y)) - t);
	elseif (Float64(-1.0 + x) <= -1.0)
		tmp = Float64(Float64(-t) - Float64(log(y) + Float64(Float64(z + -1.0) * y)));
	else
		tmp = Float64(Float64(-t) - Float64(Float64(-1.0 + x) * log(Float64(1.0 / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((-1.0 + x) <= -5000000.0)
		tmp = (x * log(y)) - t;
	elseif ((-1.0 + x) <= -1.0)
		tmp = -t - (log(y) + ((z + -1.0) * y));
	else
		tmp = -t - ((-1.0 + x) * log((1.0 / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(-1.0 + x), $MachinePrecision], -5000000.0], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(-1.0 + x), $MachinePrecision], -1.0], N[((-t) - N[(N[Log[y], $MachinePrecision] + N[(N[(z + -1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-t) - N[(N[(-1.0 + x), $MachinePrecision] * N[Log[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x 1) < -5e6

   1. Initial program 97.9%

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

   3. Taylor expanded in y around 0 99.8%

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

      1. mul-1-neg 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around inf 97.9%

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

      1. *-commutative 97.9%

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

   8. Simplified 97.9%

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

   if -5e6 < (-.f64 x 1) < -1

   1. Initial program 80.7%

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

   3. Taylor expanded in y around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

      1. flip3-- 100.0%

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

      2. associate-*l/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-define 100.0%

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

      8. metadata-eval 100.0%

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

      9. *-rgt-identity 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around 0 99.2%

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

      1. +-commutative 99.2%

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

      2. mul-1-neg 99.2%

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

      3. unsub-neg 99.2%

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

      4. mul-1-neg 99.2%

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

      5. sub-neg 99.2%

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

      6. metadata-eval 99.2%

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

      7. *-commutative 99.2%

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

      8. distribute-rgt-neg-in 99.2%

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

      9. +-commutative 99.2%

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

   10. Simplified 99.2%

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

   if -1 < (-.f64 x 1)

   1. Initial program 93.2%

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

      1. +-commutative 93.2%

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

      2. fma-define 93.2%

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

      3. sub-neg 93.2%

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

      4. metadata-eval 93.2%

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

      5. sub-neg 93.2%

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

      6. log1p-define 99.8%

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

      7. sub-neg 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 93.2%

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

      1. fma-neg 93.2%

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

      2. sub-neg 93.2%

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

      3. metadata-eval 93.2%

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

      4. +-commutative 93.2%

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

   7. Simplified 93.2%

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

   8. Taylor expanded in y around inf 93.2%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -5000000:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;-1 + x \leq -1:\\ \;\;\;\;\left(-t\right) - \left(\log y + \left(z + -1\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \left(-1 + x\right) \cdot \log \left(\frac{1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+215} \lor \neg \left(z \leq 4.2 \cdot 10^{+216}\right):\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(y + \log y \cdot \left(-1 + x\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -5.8e+215) (not (<= z 4.2e+216)))
   (- (- t) (* z y))
   (- (+ y (* (log y) (+ -1.0 x))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -5.8e+215) || !(z <= 4.2e+216)) {
		tmp = -t - (z * y);
	} else {
		tmp = (y + (log(y) * (-1.0 + x))) - 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 <= (-5.8d+215)) .or. (.not. (z <= 4.2d+216))) then
        tmp = -t - (z * y)
    else
        tmp = (y + (log(y) * ((-1.0d0) + x))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -5.8e+215) || !(z <= 4.2e+216)) {
		tmp = -t - (z * y);
	} else {
		tmp = (y + (Math.log(y) * (-1.0 + x))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -5.8e+215) or not (z <= 4.2e+216):
		tmp = -t - (z * y)
	else:
		tmp = (y + (math.log(y) * (-1.0 + x))) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -5.8e+215) || !(z <= 4.2e+216))
		tmp = Float64(Float64(-t) - Float64(z * y));
	else
		tmp = Float64(Float64(y + Float64(log(y) * Float64(-1.0 + x))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -5.8e+215) || ~((z <= 4.2e+216)))
		tmp = -t - (z * y);
	else
		tmp = (y + (log(y) * (-1.0 + x))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -5.8e+215], N[Not[LessEqual[z, 4.2e+216]], $MachinePrecision]], N[((-t) - N[(z * y), $MachinePrecision]), $MachinePrecision], N[(N[(y + N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.7999999999999999e215 or 4.20000000000000003e216 < z

   1. Initial program 57.2%

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

   3. Taylor expanded in y around 0 99.9%

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

      1. mul-1-neg 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in z around inf 77.8%

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

      1. mul-1-neg 77.8%

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

      2. *-commutative 77.8%

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

      3. distribute-rgt-neg-in 77.8%

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

   8. Simplified 77.8%

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

   if -5.7999999999999999e215 < z < 4.20000000000000003e216

   1. Initial program 95.6%

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

   3. Taylor expanded in y around 0 99.9%

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

      1. mul-1-neg 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in z around 0 95.6%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+215} \lor \neg \left(z \leq 4.2 \cdot 10^{+216}\right):\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(y + \log y \cdot \left(-1 + x\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{+217} \lor \neg \left(z \leq 4.1 \cdot 10^{+216}\right):\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -4.2e+217) (not (<= z 4.1e+216)))
   (- (- t) (* z y))
   (- (* (log y) (+ -1.0 x)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -4.2e+217) || !(z <= 4.1e+216)) {
		tmp = -t - (z * y);
	} else {
		tmp = (log(y) * (-1.0 + x)) - 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 <= (-4.2d+217)) .or. (.not. (z <= 4.1d+216))) then
        tmp = -t - (z * y)
    else
        tmp = (log(y) * ((-1.0d0) + x)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -4.2e+217) || !(z <= 4.1e+216)) {
		tmp = -t - (z * y);
	} else {
		tmp = (Math.log(y) * (-1.0 + x)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -4.2e+217) or not (z <= 4.1e+216):
		tmp = -t - (z * y)
	else:
		tmp = (math.log(y) * (-1.0 + x)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -4.2e+217) || !(z <= 4.1e+216))
		tmp = Float64(Float64(-t) - Float64(z * y));
	else
		tmp = Float64(Float64(log(y) * Float64(-1.0 + x)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -4.2e+217) || ~((z <= 4.1e+216)))
		tmp = -t - (z * y);
	else
		tmp = (log(y) * (-1.0 + x)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -4.2e+217], N[Not[LessEqual[z, 4.1e+216]], $MachinePrecision]], N[((-t) - N[(z * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.2000000000000002e217 or 4.0999999999999998e216 < z

   1. Initial program 57.2%

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

   3. Taylor expanded in y around 0 99.9%

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

      1. mul-1-neg 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in z around inf 77.8%

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

      1. mul-1-neg 77.8%

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

      2. *-commutative 77.8%

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

      3. distribute-rgt-neg-in 77.8%

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

   8. Simplified 77.8%

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

   if -4.2000000000000002e217 < z < 4.0999999999999998e216

   1. Initial program 95.6%

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

      1. +-commutative 95.6%

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

      2. fma-define 95.6%

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

      3. sub-neg 95.6%

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

      4. metadata-eval 95.6%

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

      5. sub-neg 95.6%

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

      6. log1p-define 99.9%

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

      7. sub-neg 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 95.5%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{+217} \lor \neg \left(z \leq 4.1 \cdot 10^{+216}\right):\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.65 \cdot 10^{+14} \lor \neg \left(t \leq 8.6 \cdot 10^{+74}\right):\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -2.65e+14) (not (<= t 8.6e+74)))
   (- (- t) (* z y))
   (* (log y) (+ -1.0 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -2.65e+14) || !(t <= 8.6e+74)) {
		tmp = -t - (z * y);
	} else {
		tmp = log(y) * (-1.0 + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-2.65d+14)) .or. (.not. (t <= 8.6d+74))) then
        tmp = -t - (z * y)
    else
        tmp = log(y) * ((-1.0d0) + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -2.65e+14) || !(t <= 8.6e+74)) {
		tmp = -t - (z * y);
	} else {
		tmp = Math.log(y) * (-1.0 + x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -2.65e+14) or not (t <= 8.6e+74):
		tmp = -t - (z * y)
	else:
		tmp = math.log(y) * (-1.0 + x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -2.65e+14) || !(t <= 8.6e+74))
		tmp = Float64(Float64(-t) - Float64(z * y));
	else
		tmp = Float64(log(y) * Float64(-1.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -2.65e+14) || ~((t <= 8.6e+74)))
		tmp = -t - (z * y);
	else
		tmp = log(y) * (-1.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -2.65e+14], N[Not[LessEqual[t, 8.6e+74]], $MachinePrecision]], N[((-t) - N[(z * y), $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.65e14 or 8.60000000000000001e74 < t

   1. Initial program 94.2%

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

   3. Taylor expanded in y around 0 99.9%

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

      1. mul-1-neg 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in z around inf 80.3%

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

      1. mul-1-neg 80.3%

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

      2. *-commutative 80.3%

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

      3. distribute-rgt-neg-in 80.3%

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

   8. Simplified 80.3%

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

   if -2.65e14 < t < 8.60000000000000001e74

   1. Initial program 83.4%

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

      1. +-commutative 83.4%

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

      2. fma-define 83.4%

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

      3. sub-neg 83.4%

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

      4. metadata-eval 83.4%

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

      5. sub-neg 83.4%

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

      6. log1p-define 99.9%

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

      7. sub-neg 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 83.2%

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

   6. Taylor expanded in t around 0 78.3%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.65 \cdot 10^{+14} \lor \neg \left(t \leq 8.6 \cdot 10^{+74}\right):\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -95000000 \lor \neg \left(t \leq 2400\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -95000000.0) (not (<= t 2400.0)))
   (- (* x (log y)) t)
   (* (log y) (+ -1.0 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -95000000.0) || !(t <= 2400.0)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = log(y) * (-1.0 + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-95000000.0d0)) .or. (.not. (t <= 2400.0d0))) then
        tmp = (x * log(y)) - t
    else
        tmp = log(y) * ((-1.0d0) + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -95000000.0) || !(t <= 2400.0)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = Math.log(y) * (-1.0 + x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -95000000.0) or not (t <= 2400.0):
		tmp = (x * math.log(y)) - t
	else:
		tmp = math.log(y) * (-1.0 + x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -95000000.0) || !(t <= 2400.0))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(log(y) * Float64(-1.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -95000000.0) || ~((t <= 2400.0)))
		tmp = (x * log(y)) - t;
	else
		tmp = log(y) * (-1.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -95000000.0], N[Not[LessEqual[t, 2400.0]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -9.5e7 or 2400 < t

   1. Initial program 94.3%

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

   3. Taylor expanded in y around 0 99.9%

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

      1. mul-1-neg 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around inf 93.6%

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

      1. *-commutative 93.6%

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

   8. Simplified 93.6%

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

   if -9.5e7 < t < 2400

   1. Initial program 81.6%

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

      1. +-commutative 81.6%

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

      2. fma-define 81.6%

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

      3. sub-neg 81.6%

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

      4. metadata-eval 81.6%

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

      5. sub-neg 81.6%

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

      6. log1p-define 99.9%

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

      7. sub-neg 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 81.4%

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

   6. Taylor expanded in t around 0 81.3%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -95000000 \lor \neg \left(t \leq 2400\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{+132} \lor \neg \left(z \leq 4.6 \cdot 10^{+146}\right):\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -3.9e+132) (not (<= z 4.6e+146)))
   (- (- t) (* z y))
   (- (- t) (log y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.9e+132) || !(z <= 4.6e+146)) {
		tmp = -t - (z * y);
	} else {
		tmp = -t - log(y);
	}
	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 <= (-3.9d+132)) .or. (.not. (z <= 4.6d+146))) then
        tmp = -t - (z * y)
    else
        tmp = -t - log(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.9e+132) || !(z <= 4.6e+146)) {
		tmp = -t - (z * y);
	} else {
		tmp = -t - Math.log(y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -3.9e+132) or not (z <= 4.6e+146):
		tmp = -t - (z * y)
	else:
		tmp = -t - math.log(y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -3.9e+132) || !(z <= 4.6e+146))
		tmp = Float64(Float64(-t) - Float64(z * y));
	else
		tmp = Float64(Float64(-t) - log(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -3.9e+132) || ~((z <= 4.6e+146)))
		tmp = -t - (z * y);
	else
		tmp = -t - log(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -3.9e+132], N[Not[LessEqual[z, 4.6e+146]], $MachinePrecision]], N[((-t) - N[(z * y), $MachinePrecision]), $MachinePrecision], N[((-t) - N[Log[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.90000000000000001e132 or 4.60000000000000001e146 < z

   1. Initial program 68.1%

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

   3. Taylor expanded in y around 0 99.8%

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

      1. mul-1-neg 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in z around inf 65.2%

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

      1. mul-1-neg 65.2%

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

      2. *-commutative 65.2%

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

      3. distribute-rgt-neg-in 65.2%

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

   8. Simplified 65.2%

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

   if -3.90000000000000001e132 < z < 4.60000000000000001e146

   1. Initial program 98.7%

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

      1. +-commutative 98.7%

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

      2. fma-define 98.7%

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

      3. sub-neg 98.7%

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

      4. metadata-eval 98.7%

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

      5. sub-neg 98.7%

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

      6. log1p-define 99.9%

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

      7. sub-neg 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 98.6%

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

      1. fma-neg 98.6%

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

      2. sub-neg 98.6%

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

      3. metadata-eval 98.6%

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

      4. +-commutative 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in x around 0 60.8%

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

      1. mul-1-neg 60.8%

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

   10. Simplified 60.8%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{+132} \lor \neg \left(z \leq 4.6 \cdot 10^{+146}\right):\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

3. Taylor expanded in y around 0 99.9%

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

   1. mul-1-neg 99.9%

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

5. Simplified 99.9%

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

6. Final simplification 99.9%

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

Alternative 11: 43.6% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-33} \lor \neg \left(t \leq 72000000\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -9e-33) (not (<= t 72000000.0))) (- t) (* z (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -9e-33) || !(t <= 72000000.0)) {
		tmp = -t;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-9d-33)) .or. (.not. (t <= 72000000.0d0))) then
        tmp = -t
    else
        tmp = z * -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -9e-33) || !(t <= 72000000.0)) {
		tmp = -t;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -9e-33) or not (t <= 72000000.0):
		tmp = -t
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -9e-33) || !(t <= 72000000.0))
		tmp = Float64(-t);
	else
		tmp = Float64(z * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -9e-33) || ~((t <= 72000000.0)))
		tmp = -t;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -9e-33], N[Not[LessEqual[t, 72000000.0]], $MachinePrecision]], (-t), N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -8.99999999999999982e-33 or 7.2e7 < t

   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. Step-by-step derivation

      1. +-commutative 94.5%

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

      2. fma-define 94.5%

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

      3. sub-neg 94.5%

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

      4. metadata-eval 94.5%

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

      5. sub-neg 94.5%

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

      6. log1p-define 99.9%

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

      7. sub-neg 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around inf 65.7%

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

      1. mul-1-neg 65.7%

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

   7. Simplified 65.7%

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

   if -8.99999999999999982e-33 < t < 7.2e7

   1. Initial program 80.8%

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

   3. Taylor expanded in y around 0 99.9%

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

      1. mul-1-neg 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in z around inf 22.0%

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

      1. mul-1-neg 22.0%

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

      2. *-commutative 22.0%

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

      3. distribute-rgt-neg-in 22.0%

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

   8. Simplified 22.0%

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

   9. Taylor expanded in z around inf 21.9%

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

       1. mul-1-neg 21.9%

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

       2. *-commutative 21.9%

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

       3. distribute-rgt-neg-in 21.9%

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

   11. Simplified 21.9%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-33} \lor \neg \left(t \leq 72000000\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.7% 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 88.0%

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

3. Taylor expanded in y around 0 99.9%

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

   1. mul-1-neg 99.9%

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

5. Simplified 99.9%

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

6. Taylor expanded in y around inf 47.8%

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

   1. mul-1-neg 47.8%

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

   2. distribute-rgt-neg-in 47.8%

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

   3. sub-neg 47.8%

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

   4. metadata-eval 47.8%

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

   5. +-commutative 47.8%

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

   6. distribute-neg-in 47.8%

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

   7. metadata-eval 47.8%

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

   8. unsub-neg 47.8%

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

8. Simplified 47.8%

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

9. Final simplification 47.8%

   \[\leadsto y \cdot \left(1 - z\right) - t \]
10. Add Preprocessing

Alternative 13: 46.4% accurate, 35.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

3. Taylor expanded in y around 0 99.9%

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

   1. mul-1-neg 99.9%

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

5. Simplified 99.9%

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

6. Taylor expanded in z around inf 47.6%

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

   1. mul-1-neg 47.6%

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

   2. *-commutative 47.6%

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

   3. distribute-rgt-neg-in 47.6%

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

8. Simplified 47.6%

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

9. Final simplification 47.6%

   \[\leadsto \left(-t\right) - z \cdot y \]
10. Add Preprocessing

Alternative 14: 36.5% 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 88.0%

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

   1. +-commutative 88.0%

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

   2. fma-define 88.0%

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

   3. sub-neg 88.0%

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

   4. metadata-eval 88.0%

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

   5. sub-neg 88.0%

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

   6. log1p-define 99.9%

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

   7. sub-neg 99.9%

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

   8. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in t around inf 35.7%

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

   1. mul-1-neg 35.7%

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

7. Simplified 35.7%

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

8. Final simplification 35.7%

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

Alternative 15: 2.9% accurate, 215.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

3. Taylor expanded in y around 0 99.9%

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

   1. mul-1-neg 99.9%

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

5. Simplified 99.9%

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

   1. flip3-- 65.8%

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

   2. associate-*l/ 65.8%

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

   3. metadata-eval 65.8%

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

   4. sub-neg 65.8%

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

   5. metadata-eval 65.8%

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

   6. +-commutative 65.8%

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

   7. fma-define 65.9%

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

   8. metadata-eval 65.9%

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

   9. *-rgt-identity 65.9%

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

7. Applied egg-rr 65.9%

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

8. Taylor expanded in z around 0 54.7%

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

9. Taylor expanded in y around inf 2.8%

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

10. Final simplification 2.8%

    \[\leadsto y \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(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))