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

Percentage Accurate: 89.3% → 99.8%

Time: 16.1s

Alternatives: 18

Speedup: 1.8×

• Could not converge on a ground truth

  1. x = -2.923675686302669e-266
  2. y = 1.6001596454241173e-90
  3. z = -6.10070066029329e+267
  4. t = 11313.065194933753

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lift--.f64 N/A

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

   9. *-lft-identity N/A

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

   10. fp-cancel-sub-sign-inv N/A

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

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

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

   12. *-lft-identity N/A

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

   13. lower-log1p.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower--.f64 99.8

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 66.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.136 \lor \neg \left(x \leq 34000000000\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y - 1, y, \frac{\left(\left({y}^{-1} + 0.5\right) \cdot y\right) \cdot y}{z}\right) \cdot z - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -0.136) (not (<= x 34000000000.0)))
   (* (log y) x)
   (-
    (* (fma (- (* -0.5 y) 1.0) y (/ (* (* (+ (pow y -1.0) 0.5) y) y) z)) z)
    t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -0.136) || !(x <= 34000000000.0)) {
		tmp = log(y) * x;
	} else {
		tmp = (fma(((-0.5 * y) - 1.0), y, ((((pow(y, -1.0) + 0.5) * y) * y) / z)) * z) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -0.136) || !(x <= 34000000000.0))
		tmp = Float64(log(y) * x);
	else
		tmp = Float64(Float64(fma(Float64(Float64(-0.5 * y) - 1.0), y, Float64(Float64(Float64(Float64((y ^ -1.0) + 0.5) * y) * y) / z)) * z) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -0.136], N[Not[LessEqual[x, 34000000000.0]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(N[(-0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * y + N[(N[(N[(N[(N[Power[y, -1.0], $MachinePrecision] + 0.5), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.13600000000000001 or 3.4e10 < x

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-lft-identity N/A

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

      10. fp-cancel-sub-sign-inv N/A

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

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

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

      12. *-lft-identity N/A

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

      13. lower-log1p.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower--.f64 99.6

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 74.9

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

   7. Applied rewrites 74.9%

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

   if -0.13600000000000001 < x < 3.4e10

   1. Initial program 80.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

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

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 98.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 59.7%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.136 \lor \neg \left(x \leq 34000000000\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y - 1, y, \frac{\left(\left({y}^{-1} + 0.5\right) \cdot y\right) \cdot y}{z}\right) \cdot z - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x - t\\ \mathbf{if}\;x \leq -0.136:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.22 \cdot 10^{-173}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y - 1, y, \frac{\left(\left({y}^{-1} + 0.5\right) \cdot y\right) \cdot y}{z}\right) \cdot z - t\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (log y) x) t)))
   (if (<= x -0.136)
     t_1
     (if (<= x -1.22e-173)
       (-
        (* (fma (- (* -0.5 y) 1.0) y (/ (* (* (+ (pow y -1.0) 0.5) y) y) z)) z)
        t)
       (if (<= x 1.0) (- (- (log y)) t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (log(y) * x) - t;
	double tmp;
	if (x <= -0.136) {
		tmp = t_1;
	} else if (x <= -1.22e-173) {
		tmp = (fma(((-0.5 * y) - 1.0), y, ((((pow(y, -1.0) + 0.5) * y) * y) / z)) * z) - t;
	} else if (x <= 1.0) {
		tmp = -log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(log(y) * x) - t)
	tmp = 0.0
	if (x <= -0.136)
		tmp = t_1;
	elseif (x <= -1.22e-173)
		tmp = Float64(Float64(fma(Float64(Float64(-0.5 * y) - 1.0), y, Float64(Float64(Float64(Float64((y ^ -1.0) + 0.5) * y) * y) / z)) * z) - t);
	elseif (x <= 1.0)
		tmp = Float64(Float64(-log(y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[x, -0.136], t$95$1, If[LessEqual[x, -1.22e-173], N[(N[(N[(N[(N[(-0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * y + N[(N[(N[(N[(N[Power[y, -1.0], $MachinePrecision] + 0.5), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 1.0], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.13600000000000001 or 1 < x

   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. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. log-rec N/A

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

      8. remove-double-neg N/A

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

      9. lower-log.f64 90.3

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

   5. Applied rewrites 90.3%

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

   if -0.13600000000000001 < x < -1.21999999999999993e-173

   1. Initial program 66.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

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

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 96.6%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 73.6%

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

   if -1.21999999999999993e-173 < x < 1

   1. Initial program 84.9%

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

   3. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-out-- N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. remove-double-neg N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. metadata-eval N/A

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

      10. mul-1-neg N/A

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-rgt-out N/A

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

      15. remove-double-neg N/A

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

      16. log-rec N/A

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

      17. mul-1-neg N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-+.f64 N/A

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

      21. mul-1-neg N/A

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

      22. log-rec N/A

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.3%

      \[\leadsto \left(-\log y\right) - t \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.136:\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{elif}\;x \leq -1.22 \cdot 10^{-173}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y - 1, y, \frac{\left(\left({y}^{-1} + 0.5\right) \cdot y\right) \cdot y}{z}\right) \cdot z - t\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -0.136:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.22 \cdot 10^{-173}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y - 1, y, \frac{\left(\left({y}^{-1} + 0.5\right) \cdot y\right) \cdot y}{z}\right) \cdot z - t\\ \mathbf{elif}\;x \leq 215000:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -0.136)
     t_1
     (if (<= x -1.22e-173)
       (-
        (* (fma (- (* -0.5 y) 1.0) y (/ (* (* (+ (pow y -1.0) 0.5) y) y) z)) z)
        t)
       (if (<= x 215000.0) (- (- (log y)) t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if (x <= -0.136) {
		tmp = t_1;
	} else if (x <= -1.22e-173) {
		tmp = (fma(((-0.5 * y) - 1.0), y, ((((pow(y, -1.0) + 0.5) * y) * y) / z)) * z) - t;
	} else if (x <= 215000.0) {
		tmp = -log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (x <= -0.136)
		tmp = t_1;
	elseif (x <= -1.22e-173)
		tmp = Float64(Float64(fma(Float64(Float64(-0.5 * y) - 1.0), y, Float64(Float64(Float64(Float64((y ^ -1.0) + 0.5) * y) * y) / z)) * z) - t);
	elseif (x <= 215000.0)
		tmp = Float64(Float64(-log(y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -0.136], t$95$1, If[LessEqual[x, -1.22e-173], N[(N[(N[(N[(N[(-0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * y + N[(N[(N[(N[(N[Power[y, -1.0], $MachinePrecision] + 0.5), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 215000.0], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.13600000000000001 or 215000 < x

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-lft-identity N/A

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

      10. fp-cancel-sub-sign-inv N/A

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

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

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

      12. *-lft-identity N/A

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

      13. lower-log1p.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower--.f64 99.6

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 74.3

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

   7. Applied rewrites 74.3%

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

   if -0.13600000000000001 < x < -1.21999999999999993e-173

   1. Initial program 66.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

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

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 96.6%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 73.6%

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

   if -1.21999999999999993e-173 < x < 215000

   1. Initial program 84.9%

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

   3. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-out-- N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. remove-double-neg N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. metadata-eval N/A

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

      10. mul-1-neg N/A

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-rgt-out N/A

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

      15. remove-double-neg N/A

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

      16. log-rec N/A

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

      17. mul-1-neg N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-+.f64 N/A

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

      21. mul-1-neg N/A

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

      22. log-rec N/A

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.3%

      \[\leadsto \left(-\log y\right) - t \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.136:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;x \leq -1.22 \cdot 10^{-173}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y - 1, y, \frac{\left(\left({y}^{-1} + 0.5\right) \cdot y\right) \cdot y}{z}\right) \cdot z - t\\ \mathbf{elif}\;x \leq 215000:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lift--.f64 N/A

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

   9. *-lft-identity N/A

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

   10. fp-cancel-sub-sign-inv N/A

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

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

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

   12. *-lft-identity N/A

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

   13. lower-log1p.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower--.f64 99.8

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 99.5

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

7. Applied rewrites 99.5%

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

Alternative 6: 46.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

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

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

   4. log-rec N/A

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

   5. mul-1-neg N/A

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

5. Applied rewrites 99.2%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 90.0%

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

9. Taylor expanded in y around inf

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

11. Applied rewrites 42.1%

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

12. Final simplification 42.1%

    \[\leadsto \mathsf{fma}\left(-0.5 \cdot y - 1, y, \frac{\left(\left({y}^{-1} + 0.5\right) \cdot y\right) \cdot y}{z}\right) \cdot z - t \]
13. Add Preprocessing

Alternative 7: 99.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

5. Applied rewrites 99.4%

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

Alternative 8: 99.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (((x - 1.0) * log(y)) + ((z - 1.0) * (((-0.5 * y) - 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) * ((((-0.5d0) * y) - 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) * (((-0.5 * y) - 1.0) * y))) - t;
}

Python

def code(x, y, z, t):
	return (((x - 1.0) * math.log(y)) + ((z - 1.0) * (((-0.5 * y) - 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) * Float64(Float64(Float64(-0.5 * y) - 1.0) * y))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x - 1.0) * log(y)) + ((z - 1.0) * (((-0.5 * y) - 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[(N[(N[(-0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 86.2%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 99.2

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

5. Applied rewrites 99.2%

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

6. Final simplification 99.2%

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

Alternative 9: 95.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - 1 \leq -1.1 \lor \neg \left(x - 1 \leq 200000000\right):\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-y\right) \cdot \left(z - 1\right) - \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (- x 1.0) -1.1) (not (<= (- x 1.0) 200000000.0)))
   (- (* (+ -1.0 x) (log y)) t)
   (- (- (* (- y) (- z 1.0)) (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x - 1.0) <= -1.1) || !((x - 1.0) <= 200000000.0)) {
		tmp = ((-1.0 + x) * log(y)) - t;
	} else {
		tmp = ((-y * (z - 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) <= (-1.1d0)) .or. (.not. ((x - 1.0d0) <= 200000000.0d0))) then
        tmp = (((-1.0d0) + x) * log(y)) - t
    else
        tmp = ((-y * (z - 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) <= -1.1) || !((x - 1.0) <= 200000000.0)) {
		tmp = ((-1.0 + x) * Math.log(y)) - t;
	} else {
		tmp = ((-y * (z - 1.0)) - Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x - 1.0) <= -1.1) or not ((x - 1.0) <= 200000000.0):
		tmp = ((-1.0 + x) * math.log(y)) - t
	else:
		tmp = ((-y * (z - 1.0)) - math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x - 1.0) <= -1.1) || !(Float64(x - 1.0) <= 200000000.0))
		tmp = Float64(Float64(Float64(-1.0 + x) * log(y)) - t);
	else
		tmp = Float64(Float64(Float64(Float64(-y) * Float64(z - 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) <= -1.1) || ~(((x - 1.0) <= 200000000.0)))
		tmp = ((-1.0 + x) * log(y)) - t;
	else
		tmp = ((-y * (z - 1.0)) - log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x - 1.0), $MachinePrecision], -1.1], N[Not[LessEqual[N[(x - 1.0), $MachinePrecision], 200000000.0]], $MachinePrecision]], N[(N[(N[(-1.0 + x), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[((-y) * N[(z - 1.0), $MachinePrecision]), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -1.1000000000000001 or 2e8 < (-.f64 x #s(literal 1 binary64))

   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. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-out-- N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. remove-double-neg N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. metadata-eval N/A

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

      10. mul-1-neg N/A

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-rgt-out N/A

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

      15. remove-double-neg N/A

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

      16. log-rec N/A

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

      17. mul-1-neg N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-+.f64 N/A

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

      21. mul-1-neg N/A

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

      22. log-rec N/A

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

   5. Applied rewrites 92.1%

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

   if -1.1000000000000001 < (-.f64 x #s(literal 1 binary64)) < 2e8

   1. Initial program 80.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-out-- N/A

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

      3. metadata-eval N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-out N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower--.f64 N/A

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

      12. remove-double-neg N/A

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

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

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

      14. log-rec N/A

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

      15. lower-neg.f64 N/A

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

      16. mul-1-neg N/A

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

      17. log-rec N/A

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

      18. remove-double-neg N/A

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

      19. lower-log.f64 80.0

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

   5. Applied rewrites 80.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 98.2%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - 1 \leq -1.1 \lor \neg \left(x - 1 \leq 200000000\right):\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-y\right) \cdot \left(z - 1\right) - \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

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

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

   4. log-rec N/A

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

   5. mul-1-neg N/A

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

5. Applied rewrites 99.2%

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

Alternative 11: 99.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

3. Taylor expanded in y around 0

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

   1. associate--l+ N/A

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

   2. associate-*r* N/A

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

   3. mul-1-neg N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. lower--.f64 N/A

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

   7. *-rgt-identity N/A

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

   8. fp-cancel-sub-sign-inv N/A

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

5. Applied rewrites 98.9%

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

Alternative 12: 89.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z - 1 \leq -2 \cdot 10^{+250}:\\ \;\;\;\;\left(\left(-0.5 \cdot y - 1\right) \cdot y\right) \cdot z - t\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- z 1.0) -2e+250)
   (- (* (* (- (* -0.5 y) 1.0) y) z) t)
   (- (* (+ -1.0 x) (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z - 1.0) <= -2e+250) {
		tmp = ((((-0.5 * y) - 1.0) * y) * z) - t;
	} else {
		tmp = ((-1.0 + x) * 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 - 1.0d0) <= (-2d+250)) then
        tmp = (((((-0.5d0) * y) - 1.0d0) * y) * z) - t
    else
        tmp = (((-1.0d0) + x) * 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 - 1.0) <= -2e+250) {
		tmp = ((((-0.5 * y) - 1.0) * y) * z) - t;
	} else {
		tmp = ((-1.0 + x) * Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z - 1.0) <= -2e+250:
		tmp = ((((-0.5 * y) - 1.0) * y) * z) - t
	else:
		tmp = ((-1.0 + x) * math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z - 1.0) <= -2e+250)
		tmp = Float64(Float64(Float64(Float64(Float64(-0.5 * y) - 1.0) * y) * z) - t);
	else
		tmp = Float64(Float64(Float64(-1.0 + x) * log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z - 1.0) <= -2e+250)
		tmp = ((((-0.5 * y) - 1.0) * y) * z) - t;
	else
		tmp = ((-1.0 + x) * log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z - 1.0), $MachinePrecision], -2e+250], N[(N[(N[(N[(N[(-0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(-1.0 + x), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z #s(literal 1 binary64)) < -1.9999999999999998e250

   1. Initial program 34.2%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

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

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 28.6%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 86.8%

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

   if -1.9999999999999998e250 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 88.8%

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

   3. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-out-- N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. remove-double-neg N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. metadata-eval N/A

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

      10. mul-1-neg N/A

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-rgt-out N/A

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

      15. remove-double-neg N/A

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

      16. log-rec N/A

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

      17. mul-1-neg N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-+.f64 N/A

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

      21. mul-1-neg N/A

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

      22. log-rec N/A

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

   5. Applied rewrites 87.2%

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

Alternative 13: 42.8% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{+52} \lor \neg \left(t \leq 14200\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.5 \cdot y - 1\right) \cdot y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -3e+52) (not (<= t 14200.0)))
   (- t)
   (* (* (- (* -0.5 y) 1.0) y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -3e+52) || !(t <= 14200.0)) {
		tmp = -t;
	} else {
		tmp = (((-0.5 * y) - 1.0) * y) * z;
	}
	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 <= (-3d+52)) .or. (.not. (t <= 14200.0d0))) then
        tmp = -t
    else
        tmp = ((((-0.5d0) * y) - 1.0d0) * y) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -3e+52) || !(t <= 14200.0)) {
		tmp = -t;
	} else {
		tmp = (((-0.5 * y) - 1.0) * y) * z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -3e+52) or not (t <= 14200.0):
		tmp = -t
	else:
		tmp = (((-0.5 * y) - 1.0) * y) * z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -3e+52) || !(t <= 14200.0))
		tmp = Float64(-t);
	else
		tmp = Float64(Float64(Float64(Float64(-0.5 * y) - 1.0) * y) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -3e+52) || ~((t <= 14200.0)))
		tmp = -t;
	else
		tmp = (((-0.5 * y) - 1.0) * y) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -3e+52], N[Not[LessEqual[t, 14200.0]], $MachinePrecision]], (-t), N[(N[(N[(N[(-0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3e52 or 14200 < t

   1. Initial program 94.8%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 68.8

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

   5. Applied rewrites 68.8%

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

   if -3e52 < t < 14200

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-lft-identity N/A

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

      10. fp-cancel-sub-sign-inv N/A

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

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

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

      12. *-lft-identity N/A

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

      13. lower-log1p.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower--.f64 99.7

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower--.f64 3.7

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

   7. Applied rewrites 3.7%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 20.7%

       \[\leadsto \left(\left(-0.5 \cdot y - 1\right) \cdot y\right) \cdot z \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{+52} \lor \neg \left(t \leq 14200\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.5 \cdot y - 1\right) \cdot y\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 42.8% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{+52}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 14200:\\ \;\;\;\;\left(\left(-0.5 \cdot y - 1\right) \cdot y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot z\right) \cdot -0.5 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -3e+52)
   (- t)
   (if (<= t 14200.0)
     (* (* (- (* -0.5 y) 1.0) y) z)
     (- (* (* (* y y) z) -0.5) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3e+52) {
		tmp = -t;
	} else if (t <= 14200.0) {
		tmp = (((-0.5 * y) - 1.0) * y) * z;
	} else {
		tmp = (((y * y) * z) * -0.5) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-3d+52)) then
        tmp = -t
    else if (t <= 14200.0d0) then
        tmp = ((((-0.5d0) * y) - 1.0d0) * y) * z
    else
        tmp = (((y * y) * z) * (-0.5d0)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3e+52) {
		tmp = -t;
	} else if (t <= 14200.0) {
		tmp = (((-0.5 * y) - 1.0) * y) * z;
	} else {
		tmp = (((y * y) * z) * -0.5) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -3e+52:
		tmp = -t
	elif t <= 14200.0:
		tmp = (((-0.5 * y) - 1.0) * y) * z
	else:
		tmp = (((y * y) * z) * -0.5) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -3e+52)
		tmp = Float64(-t);
	elseif (t <= 14200.0)
		tmp = Float64(Float64(Float64(Float64(-0.5 * y) - 1.0) * y) * z);
	else
		tmp = Float64(Float64(Float64(Float64(y * y) * z) * -0.5) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -3e+52)
		tmp = -t;
	elseif (t <= 14200.0)
		tmp = (((-0.5 * y) - 1.0) * y) * z;
	else
		tmp = (((y * y) * z) * -0.5) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -3e+52], (-t), If[LessEqual[t, 14200.0], N[(N[(N[(N[(-0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision], N[(N[(N[(N[(y * y), $MachinePrecision] * z), $MachinePrecision] * -0.5), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3e52

   1. Initial program 97.7%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 73.7

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

   5. Applied rewrites 73.7%

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

   if -3e52 < t < 14200

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-lft-identity N/A

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

      10. fp-cancel-sub-sign-inv N/A

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

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

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

      12. *-lft-identity N/A

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

      13. lower-log1p.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower--.f64 99.7

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower--.f64 3.7

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

   7. Applied rewrites 3.7%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 20.7%

       \[\leadsto \left(\left(-0.5 \cdot y - 1\right) \cdot y\right) \cdot z \]

   if 14200 < t

   1. Initial program 92.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

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

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 65.8%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 65.8%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{+52}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 14200:\\ \;\;\;\;\left(\left(-0.5 \cdot y - 1\right) \cdot y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot z\right) \cdot -0.5 - t\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 42.8% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{+52} \lor \neg \left(t \leq 14200\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \mathsf{fma}\left(-0.5, y, -1\right)\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -3e+52) (not (<= t 14200.0)))
   (- t)
   (* (* z (fma -0.5 y -1.0)) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -3e+52) || !(t <= 14200.0)) {
		tmp = -t;
	} else {
		tmp = (z * fma(-0.5, y, -1.0)) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -3e+52) || !(t <= 14200.0))
		tmp = Float64(-t);
	else
		tmp = Float64(Float64(z * fma(-0.5, y, -1.0)) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -3e+52], N[Not[LessEqual[t, 14200.0]], $MachinePrecision]], (-t), N[(N[(z * N[(-0.5 * y + -1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3e52 or 14200 < t

   1. Initial program 94.8%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 68.8

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

   5. Applied rewrites 68.8%

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

   if -3e52 < t < 14200

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-lft-identity N/A

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

      10. fp-cancel-sub-sign-inv N/A

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

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

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

      12. *-lft-identity N/A

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

      13. lower-log1p.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower--.f64 99.7

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower--.f64 3.7

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

   7. Applied rewrites 3.7%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 20.7%

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

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{+52} \lor \neg \left(t \leq 14200\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \mathsf{fma}\left(-0.5, y, -1\right)\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 46.6% accurate, 10.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y, z, t):
	return ((((-0.5 * y) - 1.0) * y) * z) - t

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

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

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

   4. log-rec N/A

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

   5. mul-1-neg N/A

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

5. Applied rewrites 99.2%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 29.7%

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

9. Taylor expanded in z around inf

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

11. Applied rewrites 41.6%

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

Alternative 17: 42.6% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{+52} \lor \neg \left(t \leq 14200\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -3e+52) (not (<= t 14200.0))) (- t) (* (- y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -3e+52) || !(t <= 14200.0)) {
		tmp = -t;
	} else {
		tmp = -y * z;
	}
	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 <= (-3d+52)) .or. (.not. (t <= 14200.0d0))) then
        tmp = -t
    else
        tmp = -y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -3e+52) || !(t <= 14200.0)) {
		tmp = -t;
	} else {
		tmp = -y * z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -3e+52) or not (t <= 14200.0):
		tmp = -t
	else:
		tmp = -y * z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -3e+52) || !(t <= 14200.0))
		tmp = Float64(-t);
	else
		tmp = Float64(Float64(-y) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -3e+52) || ~((t <= 14200.0)))
		tmp = -t;
	else
		tmp = -y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -3e+52], N[Not[LessEqual[t, 14200.0]], $MachinePrecision]], (-t), N[((-y) * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3e52 or 14200 < t

   1. Initial program 94.8%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 68.8

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

   5. Applied rewrites 68.8%

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

   if -3e52 < t < 14200

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-lft-identity N/A

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

      10. fp-cancel-sub-sign-inv N/A

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

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

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

      12. *-lft-identity N/A

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

      13. lower-log1p.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower--.f64 99.7

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower--.f64 3.7

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

   7. Applied rewrites 3.7%

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

   8. Taylor expanded in y around 0

      \[\leadsto \left(-1 \cdot y\right) \cdot z \]
   9. Step-by-step derivation

   10. Applied rewrites 20.5%

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

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{+52} \lor \neg \left(t \leq 14200\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 36.2% accurate, 75.3× 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 86.2%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 28.9

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

5. Applied rewrites 28.9%

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

Reproduce

herbie shell --seed 2024329 
(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))