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

Percentage Accurate: 89.1% → 99.8%

Time: 15.1s

Alternatives: 20

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift--.f64 N/A

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

   6. sub-neg N/A

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

   7. distribute-rgt-in N/A

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

   8. associate-+r+ N/A

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

   9. lower-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-log.f64 N/A

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

   14. lift--.f64 N/A

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

   15. sub-neg N/A

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

   16. lower-log1p.f64 N/A

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

   17. lower-neg.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. metadata-eval N/A

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

   20. mul-1-neg N/A

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

   21. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 85.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x - t\\ t_2 := \log \left(1 - y\right) \cdot \left(z - 1\right) + \left(x - 1\right) \cdot \log y\\ \mathbf{if}\;t\_2 \leq -2000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 339.5:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{elif}\;t\_2 \leq 700:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (log y) x) t))
        (t_2 (+ (* (log (- 1.0 y)) (- z 1.0)) (* (- x 1.0) (log y)))))
   (if (<= t_2 -2000000000.0)
     t_1
     (if (<= t_2 339.5)
       (- (* z (log1p (- y))) t)
       (if (<= t_2 700.0) (- (- (log y)) t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (log(y) * x) - t;
	double t_2 = (log((1.0 - y)) * (z - 1.0)) + ((x - 1.0) * log(y));
	double tmp;
	if (t_2 <= -2000000000.0) {
		tmp = t_1;
	} else if (t_2 <= 339.5) {
		tmp = (z * log1p(-y)) - t;
	} else if (t_2 <= 700.0) {
		tmp = -log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (Math.log(y) * x) - t;
	double t_2 = (Math.log((1.0 - y)) * (z - 1.0)) + ((x - 1.0) * Math.log(y));
	double tmp;
	if (t_2 <= -2000000000.0) {
		tmp = t_1;
	} else if (t_2 <= 339.5) {
		tmp = (z * Math.log1p(-y)) - t;
	} else if (t_2 <= 700.0) {
		tmp = -Math.log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (math.log(y) * x) - t
	t_2 = (math.log((1.0 - y)) * (z - 1.0)) + ((x - 1.0) * math.log(y))
	tmp = 0
	if t_2 <= -2000000000.0:
		tmp = t_1
	elif t_2 <= 339.5:
		tmp = (z * math.log1p(-y)) - t
	elif t_2 <= 700.0:
		tmp = -math.log(y) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(log(y) * x) - t)
	t_2 = Float64(Float64(log(Float64(1.0 - y)) * Float64(z - 1.0)) + Float64(Float64(x - 1.0) * log(y)))
	tmp = 0.0
	if (t_2 <= -2000000000.0)
		tmp = t_1;
	elseif (t_2 <= 339.5)
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	elseif (t_2 <= 700.0)
		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]}, Block[{t$95$2 = N[(N[(N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision] * N[(z - 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2000000000.0], t$95$1, If[LessEqual[t$95$2, 339.5], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$2, 700.0], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < -2e9 or 700 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y))))

   1. Initial program 95.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 92.6

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

   5. Applied rewrites 92.6%

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

   if -2e9 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 339.5

   1. Initial program 56.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-neg.f64 75.2

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

   5. Applied rewrites 75.2%

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

   if 339.5 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 700

   1. Initial program 92.7%

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

   3. Taylor expanded in x around 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-log1p.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 92.2%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 - y\right) \cdot \left(z - 1\right) + \left(x - 1\right) \cdot \log y \leq -2000000000:\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{elif}\;\log \left(1 - y\right) \cdot \left(z - 1\right) + \left(x - 1\right) \cdot \log y \leq 339.5:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{elif}\;\log \left(1 - y\right) \cdot \left(z - 1\right) + \left(x - 1\right) \cdot \log y \leq 700:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x - t\\ t_2 := \log \left(1 - y\right) \cdot \left(z - 1\right) + \left(x - 1\right) \cdot \log y\\ \mathbf{if}\;t\_2 \leq -2000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 339.5:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right), y, -1\right) \cdot y\right) \cdot z - t\\ \mathbf{elif}\;t\_2 \leq 700:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (log y) x) t))
        (t_2 (+ (* (log (- 1.0 y)) (- z 1.0)) (* (- x 1.0) (log y)))))
   (if (<= t_2 -2000000000.0)
     t_1
     (if (<= t_2 339.5)
       (-
        (* (* (fma (fma (fma -0.25 y -0.3333333333333333) y -0.5) y -1.0) y) z)
        t)
       (if (<= t_2 700.0) (- (- (log y)) t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (log(y) * x) - t;
	double t_2 = (log((1.0 - y)) * (z - 1.0)) + ((x - 1.0) * log(y));
	double tmp;
	if (t_2 <= -2000000000.0) {
		tmp = t_1;
	} else if (t_2 <= 339.5) {
		tmp = ((fma(fma(fma(-0.25, y, -0.3333333333333333), y, -0.5), y, -1.0) * y) * z) - t;
	} else if (t_2 <= 700.0) {
		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)
	t_2 = Float64(Float64(log(Float64(1.0 - y)) * Float64(z - 1.0)) + Float64(Float64(x - 1.0) * log(y)))
	tmp = 0.0
	if (t_2 <= -2000000000.0)
		tmp = t_1;
	elseif (t_2 <= 339.5)
		tmp = Float64(Float64(Float64(fma(fma(fma(-0.25, y, -0.3333333333333333), y, -0.5), y, -1.0) * y) * z) - t);
	elseif (t_2 <= 700.0)
		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]}, Block[{t$95$2 = N[(N[(N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision] * N[(z - 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2000000000.0], t$95$1, If[LessEqual[t$95$2, 339.5], N[(N[(N[(N[(N[(N[(-0.25 * y + -0.3333333333333333), $MachinePrecision] * y + -0.5), $MachinePrecision] * y + -1.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$2, 700.0], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < -2e9 or 700 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y))))

   1. Initial program 95.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 92.6

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

   5. Applied rewrites 92.6%

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

   if -2e9 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 339.5

   1. Initial program 56.5%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. lift--.f64 N/A

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

      15. sub-neg N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. mul-1-neg N/A

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

      21. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-neg.f64 75.2

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

   7. Applied rewrites 75.2%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 75.2%

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

   if 339.5 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 700

   1. Initial program 92.7%

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

   3. Taylor expanded in x around 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-log1p.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 92.2%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 - y\right) \cdot \left(z - 1\right) + \left(x - 1\right) \cdot \log y \leq -2000000000:\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{elif}\;\log \left(1 - y\right) \cdot \left(z - 1\right) + \left(x - 1\right) \cdot \log y \leq 339.5:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right), y, -1\right) \cdot y\right) \cdot z - t\\ \mathbf{elif}\;\log \left(1 - y\right) \cdot \left(z - 1\right) + \left(x - 1\right) \cdot \log y \leq 700:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ t_2 := \log \left(1 - y\right) \cdot \left(z - 1\right) + \left(x - 1\right) \cdot \log y\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+160}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 339.5:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right), y, -1\right) \cdot y\right) \cdot z - t\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+35}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x))
        (t_2 (+ (* (log (- 1.0 y)) (- z 1.0)) (* (- x 1.0) (log y)))))
   (if (<= t_2 -5e+160)
     t_1
     (if (<= t_2 339.5)
       (-
        (* (* (fma (fma (fma -0.25 y -0.3333333333333333) y -0.5) y -1.0) y) z)
        t)
       (if (<= t_2 5e+35) (- (- (log y)) t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double t_2 = (log((1.0 - y)) * (z - 1.0)) + ((x - 1.0) * log(y));
	double tmp;
	if (t_2 <= -5e+160) {
		tmp = t_1;
	} else if (t_2 <= 339.5) {
		tmp = ((fma(fma(fma(-0.25, y, -0.3333333333333333), y, -0.5), y, -1.0) * y) * z) - t;
	} else if (t_2 <= 5e+35) {
		tmp = -log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	t_2 = Float64(Float64(log(Float64(1.0 - y)) * Float64(z - 1.0)) + Float64(Float64(x - 1.0) * log(y)))
	tmp = 0.0
	if (t_2 <= -5e+160)
		tmp = t_1;
	elseif (t_2 <= 339.5)
		tmp = Float64(Float64(Float64(fma(fma(fma(-0.25, y, -0.3333333333333333), y, -0.5), y, -1.0) * y) * z) - t);
	elseif (t_2 <= 5e+35)
		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]}, Block[{t$95$2 = N[(N[(N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision] * N[(z - 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+160], t$95$1, If[LessEqual[t$95$2, 339.5], N[(N[(N[(N[(N[(N[(-0.25 * y + -0.3333333333333333), $MachinePrecision] * y + -0.5), $MachinePrecision] * y + -1.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$2, 5e+35], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < -5.0000000000000002e160 or 5.00000000000000021e35 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y))))

   1. Initial program 97.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 72.6

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

   5. Applied rewrites 72.6%

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

   if -5.0000000000000002e160 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 339.5

   1. Initial program 67.6%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. lift--.f64 N/A

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

      15. sub-neg N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. mul-1-neg N/A

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

      21. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-neg.f64 67.9

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

   7. Applied rewrites 67.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 67.9%

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

   if 339.5 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 5.00000000000000021e35

   1. Initial program 92.2%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-log1p.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 96.8

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

   5. Applied rewrites 96.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 89.0%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 - y\right) \cdot \left(z - 1\right) + \left(x - 1\right) \cdot \log y \leq -5 \cdot 10^{+160}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;\log \left(1 - y\right) \cdot \left(z - 1\right) + \left(x - 1\right) \cdot \log y \leq 339.5:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right), y, -1\right) \cdot y\right) \cdot z - t\\ \mathbf{elif}\;\log \left(1 - y\right) \cdot \left(z - 1\right) + \left(x - 1\right) \cdot \log y \leq 5 \cdot 10^{+35}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x - 1\right) \cdot \log y\\ t_2 := t\_1 - t\\ t_3 := \log \left(1 - y\right) \cdot \left(z - 1\right) + t\_1\\ \mathbf{if}\;t\_3 \leq 40:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 339.5:\\ \;\;\;\;\left(-y\right) \cdot z - t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- x 1.0) (log y)))
        (t_2 (- t_1 t))
        (t_3 (+ (* (log (- 1.0 y)) (- z 1.0)) t_1)))
   (if (<= t_3 40.0) t_2 (if (<= t_3 339.5) (- (* (- y) z) t) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - 1.0) * log(y);
	double t_2 = t_1 - t;
	double t_3 = (log((1.0 - y)) * (z - 1.0)) + t_1;
	double tmp;
	if (t_3 <= 40.0) {
		tmp = t_2;
	} else if (t_3 <= 339.5) {
		tmp = (-y * z) - t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (x - 1.0d0) * log(y)
    t_2 = t_1 - t
    t_3 = (log((1.0d0 - y)) * (z - 1.0d0)) + t_1
    if (t_3 <= 40.0d0) then
        tmp = t_2
    else if (t_3 <= 339.5d0) then
        tmp = (-y * z) - t
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - 1.0) * Math.log(y);
	double t_2 = t_1 - t;
	double t_3 = (Math.log((1.0 - y)) * (z - 1.0)) + t_1;
	double tmp;
	if (t_3 <= 40.0) {
		tmp = t_2;
	} else if (t_3 <= 339.5) {
		tmp = (-y * z) - t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - 1.0) * math.log(y)
	t_2 = t_1 - t
	t_3 = (math.log((1.0 - y)) * (z - 1.0)) + t_1
	tmp = 0
	if t_3 <= 40.0:
		tmp = t_2
	elif t_3 <= 339.5:
		tmp = (-y * z) - t
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - 1.0) * log(y))
	t_2 = Float64(t_1 - t)
	t_3 = Float64(Float64(log(Float64(1.0 - y)) * Float64(z - 1.0)) + t_1)
	tmp = 0.0
	if (t_3 <= 40.0)
		tmp = t_2;
	elseif (t_3 <= 339.5)
		tmp = Float64(Float64(Float64(-y) * z) - t);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - 1.0) * log(y);
	t_2 = t_1 - t;
	t_3 = (log((1.0 - y)) * (z - 1.0)) + t_1;
	tmp = 0.0;
	if (t_3 <= 40.0)
		tmp = t_2;
	elseif (t_3 <= 339.5)
		tmp = (-y * z) - t;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - t), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision] * N[(z - 1.0), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, 40.0], t$95$2, If[LessEqual[t$95$3, 339.5], N[(N[((-y) * z), $MachinePrecision] - t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 40 or 339.5 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y))))

   1. Initial program 94.6%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 92.8

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

   5. Applied rewrites 92.8%

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

   if 40 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 339.5

   1. Initial program 54.9%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. lift--.f64 N/A

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

      15. sub-neg N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. mul-1-neg N/A

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

      21. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-neg.f64 77.9

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

   7. Applied rewrites 77.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 77.9%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 - y\right) \cdot \left(z - 1\right) + \left(x - 1\right) \cdot \log y \leq 40:\\ \;\;\;\;\left(x - 1\right) \cdot \log y - t\\ \mathbf{elif}\;\log \left(1 - y\right) \cdot \left(z - 1\right) + \left(x - 1\right) \cdot \log y \leq 339.5:\\ \;\;\;\;\left(-y\right) \cdot z - t\\ \mathbf{else}:\\ \;\;\;\;\left(x - 1\right) \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

4. Applied rewrites 99.7%

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

Alternative 7: 99.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Final simplification 99.5%

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

Alternative 8: 99.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 99.3

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

5. Applied rewrites 99.3%

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

6. Final simplification 99.3%

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

Alternative 9: 99.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

3. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt-out N/A

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. sub-neg N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower--.f64 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower--.f64 N/A

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

   17. lower-log.f64 99.2

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

5. Applied rewrites 99.2%

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

Alternative 10: 95.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+26}:\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-15}:\\ \;\;\;\;\left(-\mathsf{fma}\left(z - 1, y, \log y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(x - 1\right) \cdot \log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.7e+26)
   (- (* (log y) x) t)
   (if (<= x 5.2e-15)
     (- (- (fma (- z 1.0) y (log y))) t)
     (- (* (- x 1.0) (log y)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.7e+26) {
		tmp = (log(y) * x) - t;
	} else if (x <= 5.2e-15) {
		tmp = -fma((z - 1.0), y, log(y)) - t;
	} else {
		tmp = ((x - 1.0) * log(y)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.7e+26)
		tmp = Float64(Float64(log(y) * x) - t);
	elseif (x <= 5.2e-15)
		tmp = Float64(Float64(-fma(Float64(z - 1.0), y, log(y))) - t);
	else
		tmp = Float64(Float64(Float64(x - 1.0) * log(y)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.7e+26], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 5.2e-15], N[((-N[(N[(z - 1.0), $MachinePrecision] * y + N[Log[y], $MachinePrecision]), $MachinePrecision]) - t), $MachinePrecision], N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.7000000000000001e26

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -1.7000000000000001e26 < x < 5.20000000000000009e-15

   1. Initial program 74.2%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-log1p.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 99.2

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

   5. Applied rewrites 99.2%

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

   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 97.8%

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

   if 5.20000000000000009e-15 < x

   1. Initial program 93.3%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 91.9

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

   5. Applied rewrites 91.9%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+26}:\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-15}:\\ \;\;\;\;\left(-\mathsf{fma}\left(z - 1, y, \log y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(x - 1\right) \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. mul-1-neg N/A

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

   5. neg-sub0 N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. associate--r+ N/A

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

   10. metadata-eval N/A

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

   11. lower--.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower--.f64 N/A

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

   15. lower-log.f64 98.8

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

5. Applied rewrites 98.8%

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

Alternative 12: 65.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{+42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+156}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right) \cdot y, y, -y\right) \cdot z - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -1.6e+42)
     t_1
     (if (<= x 1.7e+156)
       (-
        (*
         (fma (* (fma (fma -0.25 y -0.3333333333333333) y -0.5) y) y (- y))
         z)
        t)
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if (x <= -1.6e+42) {
		tmp = t_1;
	} else if (x <= 1.7e+156) {
		tmp = (fma((fma(fma(-0.25, y, -0.3333333333333333), y, -0.5) * y), y, -y) * z) - 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 <= -1.6e+42)
		tmp = t_1;
	elseif (x <= 1.7e+156)
		tmp = Float64(Float64(fma(Float64(fma(fma(-0.25, y, -0.3333333333333333), y, -0.5) * y), y, Float64(-y)) * z) - 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, -1.6e+42], t$95$1, If[LessEqual[x, 1.7e+156], N[(N[(N[(N[(N[(N[(-0.25 * y + -0.3333333333333333), $MachinePrecision] * y + -0.5), $MachinePrecision] * y), $MachinePrecision] * y + (-y)), $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.60000000000000001e42 or 1.7e156 < x

   1. Initial program 98.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 75.4

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

   5. Applied rewrites 75.4%

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

   if -1.60000000000000001e42 < x < 1.7e156

   1. Initial program 78.1%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. lift--.f64 N/A

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

      15. sub-neg N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. mul-1-neg N/A

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

      21. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-neg.f64 61.6

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

   7. Applied rewrites 61.6%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 61.2%

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

   12. Applied rewrites 61.2%

       \[\leadsto z \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right) \cdot y, y, -y\right) - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+42}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+156}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right) \cdot y, y, -y\right) \cdot z - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.9% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift--.f64 N/A

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

   6. sub-neg N/A

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

   7. distribute-rgt-in N/A

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

   8. associate-+r+ N/A

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

   9. lower-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-log.f64 N/A

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

   14. lift--.f64 N/A

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

   15. sub-neg N/A

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

   16. lower-log1p.f64 N/A

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

   17. lower-neg.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. metadata-eval N/A

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

   20. mul-1-neg N/A

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

   21. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. lower-log1p.f64 N/A

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

   4. lower-neg.f64 47.1

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

7. Applied rewrites 47.1%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 46.9%

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

12. Applied rewrites 46.9%

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

13. Final simplification 46.9%

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

Alternative 14: 46.9% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift--.f64 N/A

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

   6. sub-neg N/A

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

   7. distribute-rgt-in N/A

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

   8. associate-+r+ N/A

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

   9. lower-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-log.f64 N/A

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

   14. lift--.f64 N/A

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

   15. sub-neg N/A

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

   16. lower-log1p.f64 N/A

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

   17. lower-neg.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. metadata-eval N/A

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

   20. mul-1-neg N/A

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

   21. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. lower-log1p.f64 N/A

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

   4. lower-neg.f64 47.1

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

7. Applied rewrites 47.1%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 46.9%

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

11. Final simplification 46.9%

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

Alternative 15: 46.8% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift--.f64 N/A

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

   6. sub-neg N/A

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

   7. distribute-rgt-in N/A

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

   8. associate-+r+ N/A

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

   9. lower-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-log.f64 N/A

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

   14. lift--.f64 N/A

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

   15. sub-neg N/A

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

   16. lower-log1p.f64 N/A

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

   17. lower-neg.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. metadata-eval N/A

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

   20. mul-1-neg N/A

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

   21. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. lower-log1p.f64 N/A

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

   4. lower-neg.f64 47.1

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

7. Applied rewrites 47.1%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 46.8%

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

11. Final simplification 46.8%

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

Alternative 16: 46.8% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift--.f64 N/A

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

   6. sub-neg N/A

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

   7. distribute-rgt-in N/A

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

   8. associate-+r+ N/A

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

   9. lower-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-log.f64 N/A

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

   14. lift--.f64 N/A

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

   15. sub-neg N/A

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

   16. lower-log1p.f64 N/A

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

   17. lower-neg.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. metadata-eval N/A

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

   20. mul-1-neg N/A

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

   21. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. lower-log1p.f64 N/A

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

   4. lower-neg.f64 47.1

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

7. Applied rewrites 47.1%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 46.8%

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

11. Final simplification 46.8%

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

Alternative 17: 46.7% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift--.f64 N/A

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

   6. sub-neg N/A

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

   7. distribute-rgt-in N/A

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

   8. associate-+r+ N/A

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

   9. lower-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-log.f64 N/A

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

   14. lift--.f64 N/A

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

   15. sub-neg N/A

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

   16. lower-log1p.f64 N/A

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

   17. lower-neg.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. metadata-eval N/A

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

   20. mul-1-neg N/A

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

   21. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. lower-log1p.f64 N/A

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

   4. lower-neg.f64 47.1

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

7. Applied rewrites 47.1%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 46.7%

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

11. Final simplification 46.7%

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

Alternative 18: 46.3% accurate, 20.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift--.f64 N/A

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

   6. sub-neg N/A

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

   7. distribute-rgt-in N/A

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

   8. associate-+r+ N/A

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

   9. lower-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-log.f64 N/A

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

   14. lift--.f64 N/A

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

   15. sub-neg N/A

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

   16. lower-log1p.f64 N/A

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

   17. lower-neg.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. metadata-eval N/A

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

   20. mul-1-neg N/A

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

   21. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. lower-log1p.f64 N/A

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

   4. lower-neg.f64 47.1

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

7. Applied rewrites 47.1%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 46.4%

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

11. Final simplification 46.4%

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

Alternative 19: 35.8% 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 86.1%

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

3. Taylor expanded in 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 32.9

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

5. Applied rewrites 32.9%

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

Alternative 20: 2.2% accurate, 226.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

3. Taylor expanded in 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 32.9

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

5. Applied rewrites 32.9%

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

7. Applied rewrites 8.6%

   \[\leadsto \frac{0 - t \cdot t}{\color{blue}{0 + t}} \]
8. Step-by-step derivation

9. Applied rewrites 2.4%

   \[\leadsto t \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024277 
(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))