System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2

Percentage Accurate: 62.4% → 99.1%

Time: 19.3s

Alternatives: 18

Speedup: 11.3×

• could not determine a ground truth

  1. x = -1.7938687085899842e+171
  2. y = 5.467306047824187e-227
  3. z = 1.0271210391660805e+251
  4. t = 1.3646337956716558e-39

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot e^{z} + \left(1 - y\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(y \cdot z\right)}}\\ \mathbf{elif}\;t\_1 \leq 1.000005:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{expm1}\left(z\right) \cdot y\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (* y (exp z)) (- 1.0 y))))
   (if (<= t_1 0.0)
     (- x (/ (/ -1.0 t) (/ -1.0 (log1p (* y z)))))
     (if (<= t_1 1.000005)
       (- x (* (/ (expm1 z) t) y))
       (- x (/ (log (* (expm1 z) y)) t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * exp(z)) + (1.0 - y);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = x - ((-1.0 / t) / (-1.0 / log1p((y * z))));
	} else if (t_1 <= 1.000005) {
		tmp = x - ((expm1(z) / t) * y);
	} else {
		tmp = x - (log((expm1(z) * y)) / t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y * Math.exp(z)) + (1.0 - y);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = x - ((-1.0 / t) / (-1.0 / Math.log1p((y * z))));
	} else if (t_1 <= 1.000005) {
		tmp = x - ((Math.expm1(z) / t) * y);
	} else {
		tmp = x - (Math.log((Math.expm1(z) * y)) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y * math.exp(z)) + (1.0 - y)
	tmp = 0
	if t_1 <= 0.0:
		tmp = x - ((-1.0 / t) / (-1.0 / math.log1p((y * z))))
	elif t_1 <= 1.000005:
		tmp = x - ((math.expm1(z) / t) * y)
	else:
		tmp = x - (math.log((math.expm1(z) * y)) / t)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * exp(z)) + Float64(1.0 - y))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(x - Float64(Float64(-1.0 / t) / Float64(-1.0 / log1p(Float64(y * z)))));
	elseif (t_1 <= 1.000005)
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	else
		tmp = Float64(x - Float64(log(Float64(expm1(z) * y)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision] + N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(x - N[(N[(-1.0 / t), $MachinePrecision] / N[(-1.0 / N[Log[1 + N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.000005], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(N[(Exp[z] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))) < 0.0

   1. Initial program 2.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. inv-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 2.2

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

      10. lift-log.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. associate-+l+ N/A

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

      15. lower-log1p.f64 N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\color{blue}{\left(\mathsf{log1p}\left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right)}}^{-1}} \]

      16. +-commutative N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\color{blue}{y \cdot e^{z} + \left(\mathsf{neg}\left(y\right)\right)}\right)\right)}^{-1}} \]

      17. lift-*.f64 N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\color{blue}{y \cdot e^{z}} + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}^{-1}} \]

      18. *-commutative N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\color{blue}{e^{z} \cdot y} + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}^{-1}} \]

      19. lower-fma.f64 N/A

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

      20. lower-neg.f64 65.9

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

   4. Applied rewrites 65.9%

      \[\leadsto x - \color{blue}{\frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\mathsf{fma}\left(e^{z}, y, -y\right)\right)\right)}^{-1}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-1 N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow-1 N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 65.9

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 65.9

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

   6. Applied rewrites 65.9%

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

   7. Taylor expanded in z around 0

      \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\color{blue}{y \cdot z}\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 99.8

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

   9. Applied rewrites 99.8%

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

   if 0.0 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))) < 1.00000500000000003

   1. Initial program 78.3%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

         \[\leadsto x - \color{blue}{y \cdot \frac{e^{z} - 1}{t}} \]

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      7. lower-expm1.f64 99.5

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \cdot y \]

   5. Applied rewrites 99.5%

      \[\leadsto x - \color{blue}{\frac{\mathsf{expm1}\left(z\right)}{t} \cdot y} \]

   if 1.00000500000000003 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))

   1. Initial program 99.2%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 99.9

         \[\leadsto x - \frac{\log \left(\color{blue}{\mathsf{expm1}\left(z\right)} \cdot y\right)}{t} \]

   5. Applied rewrites 99.9%

      \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{expm1}\left(z\right) \cdot y\right)}}{t} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot e^{z} + \left(1 - y\right) \leq 0:\\ \;\;\;\;x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(y \cdot z\right)}}\\ \mathbf{elif}\;y \cdot e^{z} + \left(1 - y\right) \leq 1.000005:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{expm1}\left(z\right) \cdot y\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot e^{z} + \left(1 - y\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(y \cdot z\right)}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+136}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log 1}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (* y (exp z)) (- 1.0 y))))
   (if (<= t_1 0.0)
     (- x (/ (/ -1.0 t) (/ -1.0 (log1p (* y z)))))
     (if (<= t_1 2e+136) (- x (* (/ (expm1 z) t) y)) (- x (/ (log 1.0) t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * exp(z)) + (1.0 - y);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = x - ((-1.0 / t) / (-1.0 / log1p((y * z))));
	} else if (t_1 <= 2e+136) {
		tmp = x - ((expm1(z) / t) * y);
	} else {
		tmp = x - (log(1.0) / t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y * Math.exp(z)) + (1.0 - y);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = x - ((-1.0 / t) / (-1.0 / Math.log1p((y * z))));
	} else if (t_1 <= 2e+136) {
		tmp = x - ((Math.expm1(z) / t) * y);
	} else {
		tmp = x - (Math.log(1.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y * math.exp(z)) + (1.0 - y)
	tmp = 0
	if t_1 <= 0.0:
		tmp = x - ((-1.0 / t) / (-1.0 / math.log1p((y * z))))
	elif t_1 <= 2e+136:
		tmp = x - ((math.expm1(z) / t) * y)
	else:
		tmp = x - (math.log(1.0) / t)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * exp(z)) + Float64(1.0 - y))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(x - Float64(Float64(-1.0 / t) / Float64(-1.0 / log1p(Float64(y * z)))));
	elseif (t_1 <= 2e+136)
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	else
		tmp = Float64(x - Float64(log(1.0) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision] + N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(x - N[(N[(-1.0 / t), $MachinePrecision] / N[(-1.0 / N[Log[1 + N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+136], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[1.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))) < 0.0

   1. Initial program 2.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. inv-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 2.2

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

      10. lift-log.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. associate-+l+ N/A

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

      15. lower-log1p.f64 N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\color{blue}{\left(\mathsf{log1p}\left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right)}}^{-1}} \]

      16. +-commutative N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\color{blue}{y \cdot e^{z} + \left(\mathsf{neg}\left(y\right)\right)}\right)\right)}^{-1}} \]

      17. lift-*.f64 N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\color{blue}{y \cdot e^{z}} + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}^{-1}} \]

      18. *-commutative N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\color{blue}{e^{z} \cdot y} + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}^{-1}} \]

      19. lower-fma.f64 N/A

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

      20. lower-neg.f64 65.9

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

   4. Applied rewrites 65.9%

      \[\leadsto x - \color{blue}{\frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\mathsf{fma}\left(e^{z}, y, -y\right)\right)\right)}^{-1}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-1 N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow-1 N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 65.9

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 65.9

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

   6. Applied rewrites 65.9%

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

   7. Taylor expanded in z around 0

      \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\color{blue}{y \cdot z}\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 99.8

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

   9. Applied rewrites 99.8%

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

   if 0.0 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))) < 2.00000000000000012e136

   1. Initial program 79.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

         \[\leadsto x - \color{blue}{y \cdot \frac{e^{z} - 1}{t}} \]

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      7. lower-expm1.f64 99.0

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \cdot y \]

   5. Applied rewrites 99.0%

      \[\leadsto x - \color{blue}{\frac{\mathsf{expm1}\left(z\right)}{t} \cdot y} \]

   if 2.00000000000000012e136 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))

   1. Initial program 99.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 59.3%

      \[\leadsto x - \frac{\log \color{blue}{1}}{t} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot e^{z} + \left(1 - y\right) \leq 0:\\ \;\;\;\;x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(y \cdot z\right)}}\\ \mathbf{elif}\;y \cdot e^{z} + \left(1 - y\right) \leq 2 \cdot 10^{+136}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log 1}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 10^{-243}:\\ \;\;\;\;x - \frac{-1}{t \cdot \frac{-1}{\mathsf{log1p}\left(\mathsf{fma}\left(e^{z}, y, -y\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, y \cdot z, 0.16666666666666666 \cdot y\right), z, 0.5 \cdot y\right), z, y\right) \cdot z\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (exp z) 1e-243)
   (- x (/ -1.0 (* t (/ -1.0 (log1p (fma (exp z) y (- y)))))))
   (-
    x
    (/
     (/ -1.0 t)
     (/
      -1.0
      (log1p
       (*
        (fma
         (fma
          (fma 0.041666666666666664 (* y z) (* 0.16666666666666666 y))
          z
          (* 0.5 y))
         z
         y)
        z)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (exp(z) <= 1e-243) {
		tmp = x - (-1.0 / (t * (-1.0 / log1p(fma(exp(z), y, -y)))));
	} else {
		tmp = x - ((-1.0 / t) / (-1.0 / log1p((fma(fma(fma(0.041666666666666664, (y * z), (0.16666666666666666 * y)), z, (0.5 * y)), z, y) * z))));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (exp(z) <= 1e-243)
		tmp = Float64(x - Float64(-1.0 / Float64(t * Float64(-1.0 / log1p(fma(exp(z), y, Float64(-y)))))));
	else
		tmp = Float64(x - Float64(Float64(-1.0 / t) / Float64(-1.0 / log1p(Float64(fma(fma(fma(0.041666666666666664, Float64(y * z), Float64(0.16666666666666666 * y)), z, Float64(0.5 * y)), z, y) * z)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[Exp[z], $MachinePrecision], 1e-243], N[(x - N[(-1.0 / N[(t * N[(-1.0 / N[Log[1 + N[(N[Exp[z], $MachinePrecision] * y + (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(-1.0 / t), $MachinePrecision] / N[(-1.0 / N[Log[1 + N[(N[(N[(N[(0.041666666666666664 * N[(y * z), $MachinePrecision] + N[(0.16666666666666666 * y), $MachinePrecision]), $MachinePrecision] * z + N[(0.5 * y), $MachinePrecision]), $MachinePrecision] * z + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 9.99999999999999995e-244

   1. Initial program 79.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. inv-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 79.5

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

      10. lift-log.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. associate-+l+ N/A

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

      15. lower-log1p.f64 N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\color{blue}{\left(\mathsf{log1p}\left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right)}}^{-1}} \]

      16. +-commutative N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\color{blue}{y \cdot e^{z} + \left(\mathsf{neg}\left(y\right)\right)}\right)\right)}^{-1}} \]

      17. lift-*.f64 N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\color{blue}{y \cdot e^{z}} + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}^{-1}} \]

      18. *-commutative N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\color{blue}{e^{z} \cdot y} + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}^{-1}} \]

      19. lower-fma.f64 N/A

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

      20. lower-neg.f64 99.7

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

   4. Applied rewrites 99.7%

      \[\leadsto x - \color{blue}{\frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\mathsf{fma}\left(e^{z}, y, -y\right)\right)\right)}^{-1}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-1 N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow-1 N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 99.7

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 99.7

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

   6. Applied rewrites 99.7%

      \[\leadsto x - \color{blue}{\frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\mathsf{fma}\left(y, e^{z}, -y\right)\right)}}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-fma.f64 N/A

         \[\leadsto x - \frac{1}{\frac{-1}{\mathsf{log1p}\left(\color{blue}{y \cdot e^{z} + \left(-y\right)}\right)} \cdot \left(\mathsf{neg}\left(t\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto x - \frac{1}{\frac{-1}{\mathsf{log1p}\left(\color{blue}{e^{z} \cdot y} + \left(-y\right)\right)} \cdot \left(\mathsf{neg}\left(t\right)\right)} \]

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 99.8

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

   8. Applied rewrites 99.8%

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

   if 9.99999999999999995e-244 < (exp.f64 z)

   1. Initial program 54.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. inv-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 54.1

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

      10. lift-log.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. associate-+l+ N/A

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

      15. lower-log1p.f64 N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\color{blue}{\left(\mathsf{log1p}\left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right)}}^{-1}} \]

      16. +-commutative N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\color{blue}{y \cdot e^{z} + \left(\mathsf{neg}\left(y\right)\right)}\right)\right)}^{-1}} \]

      17. lift-*.f64 N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\color{blue}{y \cdot e^{z}} + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}^{-1}} \]

      18. *-commutative N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\color{blue}{e^{z} \cdot y} + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}^{-1}} \]

      19. lower-fma.f64 N/A

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

      20. lower-neg.f64 77.2

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

   4. Applied rewrites 77.2%

      \[\leadsto x - \color{blue}{\frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\mathsf{fma}\left(e^{z}, y, -y\right)\right)\right)}^{-1}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-1 N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow-1 N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 77.2

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 77.2

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

   6. Applied rewrites 77.2%

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

   7. Taylor expanded in z around 0

      \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\color{blue}{z \cdot \left(y + z \cdot \left(\frac{1}{2} \cdot y + z \cdot \left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right)\right)\right)}\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\color{blue}{\left(y + z \cdot \left(\frac{1}{2} \cdot y + z \cdot \left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right)\right)\right) \cdot z}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\color{blue}{\left(y + z \cdot \left(\frac{1}{2} \cdot y + z \cdot \left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right)\right)\right) \cdot z}\right)}} \]

      3. +-commutative N/A

         \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\color{blue}{\left(z \cdot \left(\frac{1}{2} \cdot y + z \cdot \left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right)\right) + y\right)} \cdot z\right)}} \]

      4. *-commutative N/A

         \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\left(\color{blue}{\left(\frac{1}{2} \cdot y + z \cdot \left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right)\right) \cdot z} + y\right) \cdot z\right)}} \]

      5. lower-fma.f64 N/A

         \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot y + z \cdot \left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right), z, y\right)} \cdot z\right)}} \]

      6. +-commutative N/A

         \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right) + \frac{1}{2} \cdot y}, z, y\right) \cdot z\right)}} \]

      7. *-commutative N/A

         \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right) \cdot z} + \frac{1}{2} \cdot y, z, y\right) \cdot z\right)}} \]

      8. lower-fma.f64 N/A

         \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y, z, \frac{1}{2} \cdot y\right)}, z, y\right) \cdot z\right)}} \]

      9. lower-fma.f64 N/A

         \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, y \cdot z, \frac{1}{6} \cdot y\right)}, z, \frac{1}{2} \cdot y\right), z, y\right) \cdot z\right)}} \]

      10. *-commutative N/A

          \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{z \cdot y}, \frac{1}{6} \cdot y\right), z, \frac{1}{2} \cdot y\right), z, y\right) \cdot z\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{z \cdot y}, \frac{1}{6} \cdot y\right), z, \frac{1}{2} \cdot y\right), z, y\right) \cdot z\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, z \cdot y, \color{blue}{\frac{1}{6} \cdot y}\right), z, \frac{1}{2} \cdot y\right), z, y\right) \cdot z\right)}} \]

      13. lower-*.f64 98.3

          \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, z \cdot y, 0.16666666666666666 \cdot y\right), z, \color{blue}{0.5 \cdot y}\right), z, y\right) \cdot z\right)}} \]

   9. Applied rewrites 98.3%

      \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, z \cdot y, 0.16666666666666666 \cdot y\right), z, 0.5 \cdot y\right), z, y\right) \cdot z}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 10^{-243}:\\ \;\;\;\;x - \frac{-1}{t \cdot \frac{-1}{\mathsf{log1p}\left(\mathsf{fma}\left(e^{z}, y, -y\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, y \cdot z, 0.16666666666666666 \cdot y\right), z, 0.5 \cdot y\right), z, y\right) \cdot z\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 10^{-243}:\\ \;\;\;\;x - \frac{\frac{-1}{t}}{\frac{\mathsf{fma}\left(-0.5, y, \frac{-1}{\mathsf{expm1}\left(z\right)}\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, y \cdot z, 0.16666666666666666 \cdot y\right), z, 0.5 \cdot y\right), z, y\right) \cdot z\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (exp z) 1e-243)
   (- x (/ (/ -1.0 t) (/ (fma -0.5 y (/ -1.0 (expm1 z))) y)))
   (-
    x
    (/
     (/ -1.0 t)
     (/
      -1.0
      (log1p
       (*
        (fma
         (fma
          (fma 0.041666666666666664 (* y z) (* 0.16666666666666666 y))
          z
          (* 0.5 y))
         z
         y)
        z)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (exp(z) <= 1e-243) {
		tmp = x - ((-1.0 / t) / (fma(-0.5, y, (-1.0 / expm1(z))) / y));
	} else {
		tmp = x - ((-1.0 / t) / (-1.0 / log1p((fma(fma(fma(0.041666666666666664, (y * z), (0.16666666666666666 * y)), z, (0.5 * y)), z, y) * z))));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (exp(z) <= 1e-243)
		tmp = Float64(x - Float64(Float64(-1.0 / t) / Float64(fma(-0.5, y, Float64(-1.0 / expm1(z))) / y)));
	else
		tmp = Float64(x - Float64(Float64(-1.0 / t) / Float64(-1.0 / log1p(Float64(fma(fma(fma(0.041666666666666664, Float64(y * z), Float64(0.16666666666666666 * y)), z, Float64(0.5 * y)), z, y) * z)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[Exp[z], $MachinePrecision], 1e-243], N[(x - N[(N[(-1.0 / t), $MachinePrecision] / N[(N[(-0.5 * y + N[(-1.0 / N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(-1.0 / t), $MachinePrecision] / N[(-1.0 / N[Log[1 + N[(N[(N[(N[(0.041666666666666664 * N[(y * z), $MachinePrecision] + N[(0.16666666666666666 * y), $MachinePrecision]), $MachinePrecision] * z + N[(0.5 * y), $MachinePrecision]), $MachinePrecision] * z + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 9.99999999999999995e-244

   1. Initial program 79.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. inv-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 79.5

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

      10. lift-log.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. associate-+l+ N/A

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

      15. lower-log1p.f64 N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\color{blue}{\left(\mathsf{log1p}\left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right)}}^{-1}} \]

      16. +-commutative N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\color{blue}{y \cdot e^{z} + \left(\mathsf{neg}\left(y\right)\right)}\right)\right)}^{-1}} \]

      17. lift-*.f64 N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\color{blue}{y \cdot e^{z}} + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}^{-1}} \]

      18. *-commutative N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\color{blue}{e^{z} \cdot y} + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}^{-1}} \]

      19. lower-fma.f64 N/A

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

      20. lower-neg.f64 99.7

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

   4. Applied rewrites 99.7%

      \[\leadsto x - \color{blue}{\frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\mathsf{fma}\left(e^{z}, y, -y\right)\right)\right)}^{-1}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-1 N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow-1 N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 99.7

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in y around 0

      \[\leadsto x - \frac{\frac{-1}{t}}{\color{blue}{\frac{\frac{-1}{2} \cdot y - \frac{1}{e^{z} - 1}}{y}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto x - \frac{\frac{-1}{t}}{\color{blue}{\frac{\frac{-1}{2} \cdot y - \frac{1}{e^{z} - 1}}{y}}} \]

      2. sub-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 92.5

         \[\leadsto x - \frac{\frac{-1}{t}}{\frac{\mathsf{fma}\left(-0.5, y, \frac{-1}{\color{blue}{\mathsf{expm1}\left(z\right)}}\right)}{y}} \]

   9. Applied rewrites 92.5%

      \[\leadsto x - \frac{\frac{-1}{t}}{\color{blue}{\frac{\mathsf{fma}\left(-0.5, y, \frac{-1}{\mathsf{expm1}\left(z\right)}\right)}{y}}} \]

   if 9.99999999999999995e-244 < (exp.f64 z)

   1. Initial program 54.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. inv-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 54.1

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

      10. lift-log.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. associate-+l+ N/A

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

      15. lower-log1p.f64 N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\color{blue}{\left(\mathsf{log1p}\left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right)}}^{-1}} \]

      16. +-commutative N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\color{blue}{y \cdot e^{z} + \left(\mathsf{neg}\left(y\right)\right)}\right)\right)}^{-1}} \]

      17. lift-*.f64 N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\color{blue}{y \cdot e^{z}} + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}^{-1}} \]

      18. *-commutative N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\color{blue}{e^{z} \cdot y} + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}^{-1}} \]

      19. lower-fma.f64 N/A

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

      20. lower-neg.f64 77.2

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

   4. Applied rewrites 77.2%

      \[\leadsto x - \color{blue}{\frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\mathsf{fma}\left(e^{z}, y, -y\right)\right)\right)}^{-1}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-1 N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow-1 N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 77.2

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 77.2

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

   6. Applied rewrites 77.2%

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

   7. Taylor expanded in z around 0

      \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\color{blue}{z \cdot \left(y + z \cdot \left(\frac{1}{2} \cdot y + z \cdot \left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right)\right)\right)}\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\color{blue}{\left(y + z \cdot \left(\frac{1}{2} \cdot y + z \cdot \left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right)\right)\right) \cdot z}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\color{blue}{\left(y + z \cdot \left(\frac{1}{2} \cdot y + z \cdot \left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right)\right)\right) \cdot z}\right)}} \]

      3. +-commutative N/A

         \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\color{blue}{\left(z \cdot \left(\frac{1}{2} \cdot y + z \cdot \left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right)\right) + y\right)} \cdot z\right)}} \]

      4. *-commutative N/A

         \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\left(\color{blue}{\left(\frac{1}{2} \cdot y + z \cdot \left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right)\right) \cdot z} + y\right) \cdot z\right)}} \]

      5. lower-fma.f64 N/A

         \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot y + z \cdot \left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right), z, y\right)} \cdot z\right)}} \]

      6. +-commutative N/A

         \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right) + \frac{1}{2} \cdot y}, z, y\right) \cdot z\right)}} \]

      7. *-commutative N/A

         \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right) \cdot z} + \frac{1}{2} \cdot y, z, y\right) \cdot z\right)}} \]

      8. lower-fma.f64 N/A

         \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y, z, \frac{1}{2} \cdot y\right)}, z, y\right) \cdot z\right)}} \]

      9. lower-fma.f64 N/A

         \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, y \cdot z, \frac{1}{6} \cdot y\right)}, z, \frac{1}{2} \cdot y\right), z, y\right) \cdot z\right)}} \]

      10. *-commutative N/A

          \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{z \cdot y}, \frac{1}{6} \cdot y\right), z, \frac{1}{2} \cdot y\right), z, y\right) \cdot z\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{z \cdot y}, \frac{1}{6} \cdot y\right), z, \frac{1}{2} \cdot y\right), z, y\right) \cdot z\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, z \cdot y, \color{blue}{\frac{1}{6} \cdot y}\right), z, \frac{1}{2} \cdot y\right), z, y\right) \cdot z\right)}} \]

      13. lower-*.f64 98.3

          \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, z \cdot y, 0.16666666666666666 \cdot y\right), z, \color{blue}{0.5 \cdot y}\right), z, y\right) \cdot z\right)}} \]

   9. Applied rewrites 98.3%

      \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, z \cdot y, 0.16666666666666666 \cdot y\right), z, 0.5 \cdot y\right), z, y\right) \cdot z}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 10^{-243}:\\ \;\;\;\;x - \frac{\frac{-1}{t}}{\frac{\mathsf{fma}\left(-0.5, y, \frac{-1}{\mathsf{expm1}\left(z\right)}\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, y \cdot z, 0.16666666666666666 \cdot y\right), z, 0.5 \cdot y\right), z, y\right) \cdot z\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot e^{z} + \left(1 - y\right) \leq 0:\\ \;\;\;\;x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(y \cdot z\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\frac{-1}{t}}{\frac{\mathsf{fma}\left(-0.5, y, \frac{-1}{\mathsf{expm1}\left(z\right)}\right)}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ (* y (exp z)) (- 1.0 y)) 0.0)
   (- x (/ (/ -1.0 t) (/ -1.0 (log1p (* y z)))))
   (- x (/ (/ -1.0 t) (/ (fma -0.5 y (/ -1.0 (expm1 z))) y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((y * exp(z)) + (1.0 - y)) <= 0.0) {
		tmp = x - ((-1.0 / t) / (-1.0 / log1p((y * z))));
	} else {
		tmp = x - ((-1.0 / t) / (fma(-0.5, y, (-1.0 / expm1(z))) / y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(y * exp(z)) + Float64(1.0 - y)) <= 0.0)
		tmp = Float64(x - Float64(Float64(-1.0 / t) / Float64(-1.0 / log1p(Float64(y * z)))));
	else
		tmp = Float64(x - Float64(Float64(-1.0 / t) / Float64(fma(-0.5, y, Float64(-1.0 / expm1(z))) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision] + N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.0], N[(x - N[(N[(-1.0 / t), $MachinePrecision] / N[(-1.0 / N[Log[1 + N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(-1.0 / t), $MachinePrecision] / N[(N[(-0.5 * y + N[(-1.0 / N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))) < 0.0

   1. Initial program 2.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. inv-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 2.2

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

      10. lift-log.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. associate-+l+ N/A

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

      15. lower-log1p.f64 N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\color{blue}{\left(\mathsf{log1p}\left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right)}}^{-1}} \]

      16. +-commutative N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\color{blue}{y \cdot e^{z} + \left(\mathsf{neg}\left(y\right)\right)}\right)\right)}^{-1}} \]

      17. lift-*.f64 N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\color{blue}{y \cdot e^{z}} + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}^{-1}} \]

      18. *-commutative N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\color{blue}{e^{z} \cdot y} + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}^{-1}} \]

      19. lower-fma.f64 N/A

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

      20. lower-neg.f64 65.9

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

   4. Applied rewrites 65.9%

      \[\leadsto x - \color{blue}{\frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\mathsf{fma}\left(e^{z}, y, -y\right)\right)\right)}^{-1}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-1 N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow-1 N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 65.9

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 65.9

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

   6. Applied rewrites 65.9%

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

   7. Taylor expanded in z around 0

      \[\leadsto x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(\color{blue}{y \cdot z}\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 99.8

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

   9. Applied rewrites 99.8%

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

   if 0.0 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))

   1. Initial program 80.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. inv-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 80.9

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

      10. lift-log.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. associate-+l+ N/A

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

      15. lower-log1p.f64 N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\color{blue}{\left(\mathsf{log1p}\left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right)}}^{-1}} \]

      16. +-commutative N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\color{blue}{y \cdot e^{z} + \left(\mathsf{neg}\left(y\right)\right)}\right)\right)}^{-1}} \]

      17. lift-*.f64 N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\color{blue}{y \cdot e^{z}} + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}^{-1}} \]

      18. *-commutative N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\color{blue}{e^{z} \cdot y} + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}^{-1}} \]

      19. lower-fma.f64 N/A

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

      20. lower-neg.f64 89.9

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

   4. Applied rewrites 89.9%

      \[\leadsto x - \color{blue}{\frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\mathsf{fma}\left(e^{z}, y, -y\right)\right)\right)}^{-1}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-1 N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow-1 N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 89.9

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 89.9

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

   6. Applied rewrites 89.9%

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

   7. Taylor expanded in y around 0

      \[\leadsto x - \frac{\frac{-1}{t}}{\color{blue}{\frac{\frac{-1}{2} \cdot y - \frac{1}{e^{z} - 1}}{y}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto x - \frac{\frac{-1}{t}}{\color{blue}{\frac{\frac{-1}{2} \cdot y - \frac{1}{e^{z} - 1}}{y}}} \]

      2. sub-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 95.0

         \[\leadsto x - \frac{\frac{-1}{t}}{\frac{\mathsf{fma}\left(-0.5, y, \frac{-1}{\color{blue}{\mathsf{expm1}\left(z\right)}}\right)}{y}} \]

   9. Applied rewrites 95.0%

      \[\leadsto x - \frac{\frac{-1}{t}}{\color{blue}{\frac{\mathsf{fma}\left(-0.5, y, \frac{-1}{\mathsf{expm1}\left(z\right)}\right)}{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot e^{z} + \left(1 - y\right) \leq 0:\\ \;\;\;\;x - \frac{\frac{-1}{t}}{\frac{-1}{\mathsf{log1p}\left(y \cdot z\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\frac{-1}{t}}{\frac{\mathsf{fma}\left(-0.5, y, \frac{-1}{\mathsf{expm1}\left(z\right)}\right)}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot e^{z} + \left(1 - y\right) \leq 2 \cdot 10^{+136}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right) \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log 1}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ (* y (exp z)) (- 1.0 y)) 2e+136)
   (- x (/ (* (expm1 z) y) t))
   (- x (/ (log 1.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((y * exp(z)) + (1.0 - y)) <= 2e+136) {
		tmp = x - ((expm1(z) * y) / t);
	} else {
		tmp = x - (log(1.0) / t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((y * Math.exp(z)) + (1.0 - y)) <= 2e+136) {
		tmp = x - ((Math.expm1(z) * y) / t);
	} else {
		tmp = x - (Math.log(1.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((y * math.exp(z)) + (1.0 - y)) <= 2e+136:
		tmp = x - ((math.expm1(z) * y) / t)
	else:
		tmp = x - (math.log(1.0) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(y * exp(z)) + Float64(1.0 - y)) <= 2e+136)
		tmp = Float64(x - Float64(Float64(expm1(z) * y) / t));
	else
		tmp = Float64(x - Float64(log(1.0) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision] + N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 2e+136], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[1.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))) < 2.00000000000000012e136

   1. Initial program 59.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 94.4

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)} \cdot y}{t} \]

   5. Applied rewrites 94.4%

      \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right) \cdot y}}{t} \]

   if 2.00000000000000012e136 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))

   1. Initial program 99.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 59.3%

      \[\leadsto x - \frac{\log \color{blue}{1}}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot e^{z} + \left(1 - y\right) \leq 2 \cdot 10^{+136}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right) \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log 1}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot e^{z} + \left(1 - y\right) \leq 2 \cdot 10^{+136}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log 1}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ (* y (exp z)) (- 1.0 y)) 2e+136)
   (- x (* (/ (expm1 z) t) y))
   (- x (/ (log 1.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((y * exp(z)) + (1.0 - y)) <= 2e+136) {
		tmp = x - ((expm1(z) / t) * y);
	} else {
		tmp = x - (log(1.0) / t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((y * Math.exp(z)) + (1.0 - y)) <= 2e+136) {
		tmp = x - ((Math.expm1(z) / t) * y);
	} else {
		tmp = x - (Math.log(1.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((y * math.exp(z)) + (1.0 - y)) <= 2e+136:
		tmp = x - ((math.expm1(z) / t) * y)
	else:
		tmp = x - (math.log(1.0) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(y * exp(z)) + Float64(1.0 - y)) <= 2e+136)
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	else
		tmp = Float64(x - Float64(log(1.0) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision] + N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 2e+136], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[1.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))) < 2.00000000000000012e136

   1. Initial program 59.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

         \[\leadsto x - \color{blue}{y \cdot \frac{e^{z} - 1}{t}} \]

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      7. lower-expm1.f64 94.0

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \cdot y \]

   5. Applied rewrites 94.0%

      \[\leadsto x - \color{blue}{\frac{\mathsf{expm1}\left(z\right)}{t} \cdot y} \]

   if 2.00000000000000012e136 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))

   1. Initial program 99.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 59.3%

      \[\leadsto x - \frac{\log \color{blue}{1}}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot e^{z} + \left(1 - y\right) \leq 2 \cdot 10^{+136}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log 1}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0.9999995:\\ \;\;\;\;x - \frac{\log 1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, z, 0.16666666666666666\right) \cdot y, z, 0.5 \cdot y\right), z, y\right) \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (exp z) 0.9999995)
   (- x (/ (log 1.0) t))
   (-
    x
    (/
     (*
      (fma
       (fma (* (fma 0.041666666666666664 z 0.16666666666666666) y) z (* 0.5 y))
       z
       y)
      z)
     t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (exp(z) <= 0.9999995) {
		tmp = x - (log(1.0) / t);
	} else {
		tmp = x - ((fma(fma((fma(0.041666666666666664, z, 0.16666666666666666) * y), z, (0.5 * y)), z, y) * z) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (exp(z) <= 0.9999995)
		tmp = Float64(x - Float64(log(1.0) / t));
	else
		tmp = Float64(x - Float64(Float64(fma(fma(Float64(fma(0.041666666666666664, z, 0.16666666666666666) * y), z, Float64(0.5 * y)), z, y) * z) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.9999995], N[(x - N[(N[Log[1.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(N[(N[(N[(0.041666666666666664 * z + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * z + N[(0.5 * y), $MachinePrecision]), $MachinePrecision] * z + y), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.999999500000000041

   1. Initial program 80.8%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 70.6%

      \[\leadsto x - \frac{\log \color{blue}{1}}{t} \]

   if 0.999999500000000041 < (exp.f64 z)

   1. Initial program 52.5%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 92.6

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)} \cdot y}{t} \]

   5. Applied rewrites 92.6%

      \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right) \cdot y}}{t} \]

   6. Taylor expanded in z around 0

      \[\leadsto x - \frac{z \cdot \color{blue}{\left(y + z \cdot \left(\frac{1}{2} \cdot y + z \cdot \left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right)\right)\right)}}{t} \]
   7. Step-by-step derivation

   8. Applied rewrites 92.6%

      \[\leadsto x - \frac{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot \mathsf{fma}\left(0.041666666666666664, z, 0.16666666666666666\right), z, 0.5 \cdot y\right), z, y\right) \cdot \color{blue}{z}}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0.9999995:\\ \;\;\;\;x - \frac{\log 1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, z, 0.16666666666666666\right) \cdot y, z, 0.5 \cdot y\right), z, y\right) \cdot z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 10^{-243}:\\ \;\;\;\;x - \frac{y}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, z, 0.16666666666666666\right) \cdot y, z, 0.5 \cdot y\right), z, y\right) \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (exp z) 1e-243)
   (- x (* (/ y t) z))
   (-
    x
    (/
     (*
      (fma
       (fma (* (fma 0.041666666666666664 z 0.16666666666666666) y) z (* 0.5 y))
       z
       y)
      z)
     t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (exp(z) <= 1e-243) {
		tmp = x - ((y / t) * z);
	} else {
		tmp = x - ((fma(fma((fma(0.041666666666666664, z, 0.16666666666666666) * y), z, (0.5 * y)), z, y) * z) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (exp(z) <= 1e-243)
		tmp = Float64(x - Float64(Float64(y / t) * z));
	else
		tmp = Float64(x - Float64(Float64(fma(fma(Float64(fma(0.041666666666666664, z, 0.16666666666666666) * y), z, Float64(0.5 * y)), z, y) * z) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[Exp[z], $MachinePrecision], 1e-243], N[(x - N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(N[(N[(N[(0.041666666666666664 * z + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * z + N[(0.5 * y), $MachinePrecision]), $MachinePrecision] * z + y), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 9.99999999999999995e-244

   1. Initial program 79.5%

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

   3. Taylor expanded in z around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 49.3

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

   5. Applied rewrites 49.3%

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

   if 9.99999999999999995e-244 < (exp.f64 z)

   1. Initial program 54.1%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 91.7

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)} \cdot y}{t} \]

   5. Applied rewrites 91.7%

      \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right) \cdot y}}{t} \]

   6. Taylor expanded in z around 0

      \[\leadsto x - \frac{z \cdot \color{blue}{\left(y + z \cdot \left(\frac{1}{2} \cdot y + z \cdot \left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right)\right)\right)}}{t} \]
   7. Step-by-step derivation

   8. Applied rewrites 91.7%

      \[\leadsto x - \frac{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot \mathsf{fma}\left(0.041666666666666664, z, 0.16666666666666666\right), z, 0.5 \cdot y\right), z, y\right) \cdot \color{blue}{z}}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 10^{-243}:\\ \;\;\;\;x - \frac{y}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, z, 0.16666666666666666\right) \cdot y, z, 0.5 \cdot y\right), z, y\right) \cdot z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 10^{-243}:\\ \;\;\;\;x - \frac{y}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z, 0.5\right), z, 1\right) \cdot y\right) \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (exp z) 1e-243)
   (- x (* (/ y t) z))
   (- x (/ (* (* (fma (fma 0.16666666666666666 z 0.5) z 1.0) y) z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (exp(z) <= 1e-243) {
		tmp = x - ((y / t) * z);
	} else {
		tmp = x - (((fma(fma(0.16666666666666666, z, 0.5), z, 1.0) * y) * z) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (exp(z) <= 1e-243)
		tmp = Float64(x - Float64(Float64(y / t) * z));
	else
		tmp = Float64(x - Float64(Float64(Float64(fma(fma(0.16666666666666666, z, 0.5), z, 1.0) * y) * z) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[Exp[z], $MachinePrecision], 1e-243], N[(x - N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(N[(N[(0.16666666666666666 * z + 0.5), $MachinePrecision] * z + 1.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 9.99999999999999995e-244

   1. Initial program 79.5%

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

   3. Taylor expanded in z around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 49.3

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

   5. Applied rewrites 49.3%

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

   if 9.99999999999999995e-244 < (exp.f64 z)

   1. Initial program 54.1%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

         \[\leadsto x - \frac{\color{blue}{\left(y + z \cdot \left(\frac{1}{6} \cdot \left(z \cdot \left(y + \left(-3 \cdot {y}^{2} + 2 \cdot {y}^{3}\right)\right)\right) + \frac{1}{2} \cdot \left(y + -1 \cdot {y}^{2}\right)\right)\right) \cdot z}}{t} \]

      2. lower-*.f64 N/A

         \[\leadsto x - \frac{\color{blue}{\left(y + z \cdot \left(\frac{1}{6} \cdot \left(z \cdot \left(y + \left(-3 \cdot {y}^{2} + 2 \cdot {y}^{3}\right)\right)\right) + \frac{1}{2} \cdot \left(y + -1 \cdot {y}^{2}\right)\right)\right) \cdot z}}{t} \]

   5. Applied rewrites 71.3%

      \[\leadsto x - \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot z, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(2, y, -3\right), y\right), \left(y - y \cdot y\right) \cdot 0.5\right), z, y\right) \cdot z}}{t} \]

   6. Taylor expanded in y around 0

      \[\leadsto x - \frac{\left(y \cdot \left(1 + z \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot z\right)\right)\right) \cdot z}{t} \]
   7. Step-by-step derivation

   8. Applied rewrites 91.7%

      \[\leadsto x - \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z, 0.5\right), z, 1\right) \cdot y\right) \cdot z}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 87.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{+134}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(y \cdot z, 0.5, y\right) \cdot z + 1\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 4.3e+134)
   (- x (* (/ (expm1 z) t) y))
   (- x (/ (log (+ (* (fma (* y z) 0.5 y) z) 1.0)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.3e+134) {
		tmp = x - ((expm1(z) / t) * y);
	} else {
		tmp = x - (log(((fma((y * z), 0.5, y) * z) + 1.0)) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 4.3e+134)
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	else
		tmp = Float64(x - Float64(log(Float64(Float64(fma(Float64(y * z), 0.5, y) * z) + 1.0)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 4.3e+134], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(N[(N[(N[(y * z), $MachinePrecision] * 0.5 + y), $MachinePrecision] * z), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.3e134

   1. Initial program 67.2%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

         \[\leadsto x - \color{blue}{y \cdot \frac{e^{z} - 1}{t}} \]

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      7. lower-expm1.f64 91.9

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \cdot y \]

   5. Applied rewrites 91.9%

      \[\leadsto x - \color{blue}{\frac{\mathsf{expm1}\left(z\right)}{t} \cdot y} \]

   if 4.3e134 < y

   1. Initial program 6.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 92.3

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

   5. Applied rewrites 92.3%

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

   7. Applied rewrites 92.4%

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

Alternative 12: 87.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{+134}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, y \cdot z, y\right), z, 1\right)\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 4.3e+134)
   (- x (* (/ (expm1 z) t) y))
   (- x (/ (log (fma (fma 0.5 (* y z) y) z 1.0)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.3e+134) {
		tmp = x - ((expm1(z) / t) * y);
	} else {
		tmp = x - (log(fma(fma(0.5, (y * z), y), z, 1.0)) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 4.3e+134)
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	else
		tmp = Float64(x - Float64(log(fma(fma(0.5, Float64(y * z), y), z, 1.0)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 4.3e+134], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(N[(0.5 * N[(y * z), $MachinePrecision] + y), $MachinePrecision] * z + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.3e134

   1. Initial program 67.2%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

         \[\leadsto x - \color{blue}{y \cdot \frac{e^{z} - 1}{t}} \]

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      7. lower-expm1.f64 91.9

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \cdot y \]

   5. Applied rewrites 91.9%

      \[\leadsto x - \color{blue}{\frac{\mathsf{expm1}\left(z\right)}{t} \cdot y} \]

   if 4.3e134 < y

   1. Initial program 6.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 92.3

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

   5. Applied rewrites 92.3%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{+134}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, y \cdot z, y\right), z, 1\right)\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 10^{-243}:\\ \;\;\;\;x - \frac{y}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(\mathsf{fma}\left(0.5, z, 1\right) \cdot z\right) \cdot y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (exp z) 1e-243)
   (- x (* (/ y t) z))
   (- x (/ (* (* (fma 0.5 z 1.0) z) y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (exp(z) <= 1e-243) {
		tmp = x - ((y / t) * z);
	} else {
		tmp = x - (((fma(0.5, z, 1.0) * z) * y) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (exp(z) <= 1e-243)
		tmp = Float64(x - Float64(Float64(y / t) * z));
	else
		tmp = Float64(x - Float64(Float64(Float64(fma(0.5, z, 1.0) * z) * y) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[Exp[z], $MachinePrecision], 1e-243], N[(x - N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(N[(0.5 * z + 1.0), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 9.99999999999999995e-244

   1. Initial program 79.5%

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

   3. Taylor expanded in z around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 49.3

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

   5. Applied rewrites 49.3%

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

   if 9.99999999999999995e-244 < (exp.f64 z)

   1. Initial program 54.1%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 91.7

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)} \cdot y}{t} \]

   5. Applied rewrites 91.7%

      \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right) \cdot y}}{t} \]

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 91.6%

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

Alternative 14: 75.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot e^{z} + \left(1 - y\right) \leq 0:\\ \;\;\;\;x - \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{t} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ (* y (exp z)) (- 1.0 y)) 0.0)
   (- x (/ (* y z) t))
   (- x (* (/ y t) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((y * exp(z)) + (1.0 - y)) <= 0.0) {
		tmp = x - ((y * z) / t);
	} else {
		tmp = x - ((y / t) * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((y * exp(z)) + (1.0d0 - y)) <= 0.0d0) then
        tmp = x - ((y * z) / t)
    else
        tmp = x - ((y / t) * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((y * Math.exp(z)) + (1.0 - y)) <= 0.0) {
		tmp = x - ((y * z) / t);
	} else {
		tmp = x - ((y / t) * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((y * math.exp(z)) + (1.0 - y)) <= 0.0:
		tmp = x - ((y * z) / t)
	else:
		tmp = x - ((y / t) * z)
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))) < 0.0

   1. Initial program 2.2%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 82.2

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

   5. Applied rewrites 82.2%

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

   if 0.0 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))

   1. Initial program 80.9%

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

   3. Taylor expanded in z around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 77.5

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

   5. Applied rewrites 77.5%

      \[\leadsto x - \color{blue}{\frac{y}{t} \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot e^{z} + \left(1 - y\right) \leq 0:\\ \;\;\;\;x - \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{t} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 87.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{+134}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(z, y, 1\right)\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 4.3e+134)
   (- x (* (/ (expm1 z) t) y))
   (- x (/ (log (fma z y 1.0)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.3e+134) {
		tmp = x - ((expm1(z) / t) * y);
	} else {
		tmp = x - (log(fma(z, y, 1.0)) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 4.3e+134)
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	else
		tmp = Float64(x - Float64(log(fma(z, y, 1.0)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 4.3e+134], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(z * y + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.3e134

   1. Initial program 67.2%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

         \[\leadsto x - \color{blue}{y \cdot \frac{e^{z} - 1}{t}} \]

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      7. lower-expm1.f64 91.9

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \cdot y \]

   5. Applied rewrites 91.9%

      \[\leadsto x - \color{blue}{\frac{\mathsf{expm1}\left(z\right)}{t} \cdot y} \]

   if 4.3e134 < y

   1. Initial program 6.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 92.3

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

   5. Applied rewrites 92.3%

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

Alternative 16: 79.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -760:\\ \;\;\;\;x - \frac{\frac{-1}{t}}{\frac{\mathsf{fma}\left(z, \frac{\mathsf{fma}\left(-0.5, y, 0.5\right) \cdot y}{y \cdot y}, \frac{-1}{y}\right)}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, z, 0.16666666666666666\right) \cdot y, z, 0.5 \cdot y\right), z, y\right) \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -760.0)
   (-
    x
    (/ (/ -1.0 t) (/ (fma z (/ (* (fma -0.5 y 0.5) y) (* y y)) (/ -1.0 y)) z)))
   (-
    x
    (/
     (*
      (fma
       (fma (* (fma 0.041666666666666664 z 0.16666666666666666) y) z (* 0.5 y))
       z
       y)
      z)
     t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -760.0) {
		tmp = x - ((-1.0 / t) / (fma(z, ((fma(-0.5, y, 0.5) * y) / (y * y)), (-1.0 / y)) / z));
	} else {
		tmp = x - ((fma(fma((fma(0.041666666666666664, z, 0.16666666666666666) * y), z, (0.5 * y)), z, y) * z) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -760.0)
		tmp = Float64(x - Float64(Float64(-1.0 / t) / Float64(fma(z, Float64(Float64(fma(-0.5, y, 0.5) * y) / Float64(y * y)), Float64(-1.0 / y)) / z)));
	else
		tmp = Float64(x - Float64(Float64(fma(fma(Float64(fma(0.041666666666666664, z, 0.16666666666666666) * y), z, Float64(0.5 * y)), z, y) * z) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -760.0], N[(x - N[(N[(-1.0 / t), $MachinePrecision] / N[(N[(z * N[(N[(N[(-0.5 * y + 0.5), $MachinePrecision] * y), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(N[(N[(N[(0.041666666666666664 * z + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * z + N[(0.5 * y), $MachinePrecision]), $MachinePrecision] * z + y), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -760

   1. Initial program 79.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. inv-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 79.2

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

      10. lift-log.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. associate-+l+ N/A

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

      15. lower-log1p.f64 N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\color{blue}{\left(\mathsf{log1p}\left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right)}}^{-1}} \]

      16. +-commutative N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\color{blue}{y \cdot e^{z} + \left(\mathsf{neg}\left(y\right)\right)}\right)\right)}^{-1}} \]

      17. lift-*.f64 N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\color{blue}{y \cdot e^{z}} + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}^{-1}} \]

      18. *-commutative N/A

          \[\leadsto x - \frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\color{blue}{e^{z} \cdot y} + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}^{-1}} \]

      19. lower-fma.f64 N/A

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

      20. lower-neg.f64 99.7

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

   4. Applied rewrites 99.7%

      \[\leadsto x - \color{blue}{\frac{{t}^{-1}}{{\left(\mathsf{log1p}\left(\mathsf{fma}\left(e^{z}, y, -y\right)\right)\right)}^{-1}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-1 N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow-1 N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 99.7

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 63.8%

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

   if -760 < z

   1. Initial program 54.3%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 91.3

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)} \cdot y}{t} \]

   5. Applied rewrites 91.3%

      \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right) \cdot y}}{t} \]

   6. Taylor expanded in z around 0

      \[\leadsto x - \frac{z \cdot \color{blue}{\left(y + z \cdot \left(\frac{1}{2} \cdot y + z \cdot \left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right)\right)\right)}}{t} \]
   7. Step-by-step derivation

   8. Applied rewrites 91.2%

      \[\leadsto x - \frac{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot \mathsf{fma}\left(0.041666666666666664, z, 0.16666666666666666\right), z, 0.5 \cdot y\right), z, y\right) \cdot \color{blue}{z}}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -760:\\ \;\;\;\;x - \frac{\frac{-1}{t}}{\frac{\mathsf{fma}\left(z, \frac{\mathsf{fma}\left(-0.5, y, 0.5\right) \cdot y}{y \cdot y}, \frac{-1}{y}\right)}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, z, 0.16666666666666666\right) \cdot y, z, 0.5 \cdot y\right), z, y\right) \cdot z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 75.8% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-51}:\\ \;\;\;\;x - \frac{y}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.5e-51) (- x (* (/ y t) z)) (- x (* (/ z t) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.5e-51) {
		tmp = x - ((y / t) * z);
	} else {
		tmp = x - ((z / t) * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1.5d-51)) then
        tmp = x - ((y / t) * z)
    else
        tmp = x - ((z / t) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.5e-51) {
		tmp = x - ((y / t) * z);
	} else {
		tmp = x - ((z / t) * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.5e-51:
		tmp = x - ((y / t) * z)
	else:
		tmp = x - ((z / t) * y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.5e-51)
		tmp = Float64(x - Float64(Float64(y / t) * z));
	else
		tmp = Float64(x - Float64(Float64(z / t) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.5e-51)
		tmp = x - ((y / t) * z);
	else
		tmp = x - ((z / t) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.5e-51], N[(x - N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.50000000000000001e-51

   1. Initial program 76.6%

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

   3. Taylor expanded in z around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 53.8

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

   5. Applied rewrites 53.8%

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

   if -1.50000000000000001e-51 < z

   1. Initial program 53.4%

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

   3. Taylor expanded in z around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 86.8

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

   5. Applied rewrites 86.8%

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

   7. Applied rewrites 91.9%

      \[\leadsto x - y \cdot \color{blue}{\frac{z}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-51}:\\ \;\;\;\;x - \frac{y}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{t} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 74.4% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ x - \frac{z}{t} \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.8%

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

3. Taylor expanded in z around 0

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 74.8

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

5. Applied rewrites 74.8%

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

7. Applied rewrites 76.2%

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

8. Final simplification 76.2%

   \[\leadsto x - \frac{z}{t} \cdot y \]
9. Add Preprocessing

Developer Target 1: 75.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-0.5}{y \cdot t}\\ \mathbf{if}\;z < -2.8874623088207947 \cdot 10^{+119}:\\ \;\;\;\;\left(x - \frac{t\_1}{z \cdot z}\right) - t\_1 \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- 0.5) (* y t))))
   (if (< z -2.8874623088207947e+119)
     (- (- x (/ t_1 (* z z))) (* t_1 (/ (/ 2.0 z) (* z z))))
     (- x (/ (log (+ 1.0 (* z y))) t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -0.5 / (y * t);
	double tmp;
	if (z < -2.8874623088207947e+119) {
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)));
	} else {
		tmp = x - (log((1.0 + (z * y))) / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -0.5d0 / (y * t)
    if (z < (-2.8874623088207947d+119)) then
        tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0d0 / z) / (z * z)))
    else
        tmp = x - (log((1.0d0 + (z * y))) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = -0.5 / (y * t);
	double tmp;
	if (z < -2.8874623088207947e+119) {
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)));
	} else {
		tmp = x - (Math.log((1.0 + (z * y))) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -0.5 / (y * t)
	tmp = 0
	if z < -2.8874623088207947e+119:
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)))
	else:
		tmp = x - (math.log((1.0 + (z * y))) / t)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-0.5) / Float64(y * t))
	tmp = 0.0
	if (z < -2.8874623088207947e+119)
		tmp = Float64(Float64(x - Float64(t_1 / Float64(z * z))) - Float64(t_1 * Float64(Float64(2.0 / z) / Float64(z * z))));
	else
		tmp = Float64(x - Float64(log(Float64(1.0 + Float64(z * y))) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -0.5 / (y * t);
	tmp = 0.0;
	if (z < -2.8874623088207947e+119)
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)));
	else
		tmp = x - (log((1.0 + (z * y))) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-0.5) / N[(y * t), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -2.8874623088207947e+119], N[(N[(x - N[(t$95$1 / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[(N[(2.0 / z), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(1.0 + N[(z * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024277 
(FPCore (x y z t)
  :name "System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -288746230882079470000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (- x (/ (/ (- 1/2) (* y t)) (* z z))) (* (/ (- 1/2) (* y t)) (/ (/ 2 z) (* z z)))) (- x (/ (log (+ 1 (* z y))) t))))

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