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

Percentage Accurate: 61.9% → 99.3%

Time: 18.6s

Alternatives: 14

Speedup: 11.3×

• could not determine a ground truth

  1. x = 1.187658709777295e-10
  2. y = 2.5866219074624454e+155
  3. z = 266892014087392.97
  4. t = -6.395461947135477e+153

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: 61.9% 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: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{z} \cdot y + \left(1 - y\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{t}, \mathsf{log1p}\left(y \cdot z\right), x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;x - \frac{1}{\mathsf{fma}\left(0.5, t, \frac{\frac{t}{\mathsf{expm1}\left(z\right)}}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (* (exp z) y) (- 1.0 y))))
   (if (<= t_1 0.0)
     (fma (/ -1.0 t) (log1p (* y z)) x)
     (if (<= t_1 2.0)
       (- x (/ 1.0 (fma 0.5 t (/ (/ t (expm1 z)) y))))
       (- x (/ (log (* y (expm1 z))) t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (exp(z) * y) + (1.0 - y);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = fma((-1.0 / t), log1p((y * z)), x);
	} else if (t_1 <= 2.0) {
		tmp = x - (1.0 / fma(0.5, t, ((t / expm1(z)) / y)));
	} else {
		tmp = x - (log((y * expm1(z))) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(exp(z) * y) + Float64(1.0 - y))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = fma(Float64(-1.0 / t), log1p(Float64(y * z)), x);
	elseif (t_1 <= 2.0)
		tmp = Float64(x - Float64(1.0 / fma(0.5, t, Float64(Float64(t / expm1(z)) / y))));
	else
		tmp = Float64(x - Float64(log(Float64(y * expm1(z))) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Exp[z], $MachinePrecision] * y), $MachinePrecision] + N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[(-1.0 / t), $MachinePrecision] * N[Log[1 + N[(y * z), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(x - N[(1.0 / N[(0.5 * t + N[(N[(t / N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(y * N[(Exp[z] - 1), $MachinePrecision]), $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 2.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 60.5%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

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

   1. Initial program 76.7%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 73.1

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

   5. Applied rewrites 73.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 73.1

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

   7. Applied rewrites 73.1%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 99.9

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

   10. Applied rewrites 99.9%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 99.9%

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

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

   1. Initial program 99.6%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 97.7

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

   5. Applied rewrites 97.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \cdot y + \left(1 - y\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{t}, \mathsf{log1p}\left(y \cdot z\right), x\right)\\ \mathbf{elif}\;e^{z} \cdot y + \left(1 - y\right) \leq 2:\\ \;\;\;\;x - \frac{1}{\mathsf{fma}\left(0.5, t, \frac{\frac{t}{\mathsf{expm1}\left(z\right)}}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{z} \cdot y + \left(1 - y\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{t}, \mathsf{log1p}\left(y \cdot z\right), x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+67}:\\ \;\;\;\;x - \frac{1}{\mathsf{fma}\left(0.5, t, \frac{\frac{t}{\mathsf{expm1}\left(z\right)}}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{-t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (* (exp z) y) (- 1.0 y))))
   (if (<= t_1 0.0)
     (fma (/ -1.0 t) (log1p (* y z)) x)
     (if (<= t_1 5e+67)
       (- x (/ 1.0 (fma 0.5 t (/ (/ t (expm1 z)) y))))
       (/ (log1p (* y (expm1 z))) (- t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (exp(z) * y) + (1.0 - y);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = fma((-1.0 / t), log1p((y * z)), x);
	} else if (t_1 <= 5e+67) {
		tmp = x - (1.0 / fma(0.5, t, ((t / expm1(z)) / y)));
	} else {
		tmp = log1p((y * expm1(z))) / -t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(exp(z) * y) + Float64(1.0 - y))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = fma(Float64(-1.0 / t), log1p(Float64(y * z)), x);
	elseif (t_1 <= 5e+67)
		tmp = Float64(x - Float64(1.0 / fma(0.5, t, Float64(Float64(t / expm1(z)) / y))));
	else
		tmp = Float64(log1p(Float64(y * expm1(z))) / Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Exp[z], $MachinePrecision] * y), $MachinePrecision] + N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[(-1.0 / t), $MachinePrecision] * N[Log[1 + N[(y * z), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+67], N[(x - N[(1.0 / N[(0.5 * t + N[(N[(t / N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + N[(y * N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t)), $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 2.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 60.5%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if 0.0 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))) < 4.99999999999999976e67

   1. Initial program 77.6%

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

   3. Taylor expanded in z around 0

      \[\leadsto x - \frac{\log \color{blue}{\left(1 + y \cdot z\right)}}{t} \]
   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 72.5

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

   5. Applied rewrites 72.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 72.5

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

   7. Applied rewrites 72.5%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 98.8

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

   10. Applied rewrites 98.8%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 98.8%

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

   if 4.99999999999999976e67 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))

   1. Initial program 99.6%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. associate-+l+ N/A

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

      6. *-rgt-identity N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. distribute-lft-out-- N/A

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

      9. lower-log1p.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-expm1.f64 N/A

          \[\leadsto \frac{\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(z\right)} \cdot y\right)}{\mathsf{neg}\left(t\right)} \]

      13. lower-neg.f64 69.1

          \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(z\right) \cdot y\right)}{\color{blue}{-t}} \]

   5. Applied rewrites 69.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(z\right) \cdot y\right)}{-t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \cdot y + \left(1 - y\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{t}, \mathsf{log1p}\left(y \cdot z\right), x\right)\\ \mathbf{elif}\;e^{z} \cdot y + \left(1 - y\right) \leq 5 \cdot 10^{+67}:\\ \;\;\;\;x - \frac{1}{\mathsf{fma}\left(0.5, t, \frac{\frac{t}{\mathsf{expm1}\left(z\right)}}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{-t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{z} \cdot y + \left(1 - y\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{t}, \mathsf{log1p}\left(y \cdot z\right), x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{0.5 \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (* (exp z) y) (- 1.0 y))))
   (if (<= t_1 0.0)
     (fma (/ -1.0 t) (log1p (* y z)) x)
     (if (<= t_1 2.0) (- x (* (/ (expm1 z) t) y)) (- x (/ 1.0 (* 0.5 t)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (exp(z) * y) + (1.0 - y);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = fma((-1.0 / t), log1p((y * z)), x);
	} else if (t_1 <= 2.0) {
		tmp = x - ((expm1(z) / t) * y);
	} else {
		tmp = x - (1.0 / (0.5 * t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(exp(z) * y) + Float64(1.0 - y))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = fma(Float64(-1.0 / t), log1p(Float64(y * z)), x);
	elseif (t_1 <= 2.0)
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	else
		tmp = Float64(x - Float64(1.0 / Float64(0.5 * t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Exp[z], $MachinePrecision] * y), $MachinePrecision] + N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[(-1.0 / t), $MachinePrecision] * N[Log[1 + N[(y * z), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x - N[(1.0 / N[(0.5 * t), $MachinePrecision]), $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 2.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 60.5%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

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

   1. Initial program 76.7%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 99.7

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

   5. Applied rewrites 99.7%

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

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

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

      \[\leadsto x - \frac{\log \color{blue}{\left(1 + y \cdot z\right)}}{t} \]
   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 30.2

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

   5. Applied rewrites 30.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 30.2

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

   7. Applied rewrites 30.2%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 44.5

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

   10. Applied rewrites 44.5%

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

   11. Taylor expanded in y around inf

       \[\leadsto x - \frac{1}{\frac{1}{2} \cdot \color{blue}{t}} \]
   12. Step-by-step derivation

   13. Applied rewrites 45.8%

       \[\leadsto x - \frac{1}{0.5 \cdot \color{blue}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \cdot y + \left(1 - y\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{t}, \mathsf{log1p}\left(y \cdot z\right), x\right)\\ \mathbf{elif}\;e^{z} \cdot y + \left(1 - y\right) \leq 2:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{0.5 \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(N[Exp[z], $MachinePrecision] * y), $MachinePrecision] + N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(-1.0 / t), $MachinePrecision] * N[Log[1 + N[(y * z), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision], N[(x - N[(1.0 / N[(0.5 * t + N[(N[(t / N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $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 2.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 60.5%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

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

   1. Initial program 80.5%

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

   3. Taylor expanded in z around 0

      \[\leadsto x - \frac{\log \color{blue}{\left(1 + y \cdot z\right)}}{t} \]
   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 66.0

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

   5. Applied rewrites 66.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 66.0

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

   7. Applied rewrites 66.0%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 90.7

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

   10. Applied rewrites 90.7%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 90.9%

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

4. Final simplification 93.5%

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

Alternative 5: 88.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \cdot y + \left(1 - y\right) \leq 2:\\ \;\;\;\;x - \frac{y \cdot \mathsf{expm1}\left(z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{0.5 \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ (* (exp z) y) (- 1.0 y)) 2.0)
   (- x (/ (* y (expm1 z)) t))
   (- x (/ 1.0 (* 0.5 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((exp(z) * y) + (1.0 - y)) <= 2.0) {
		tmp = x - ((y * expm1(z)) / t);
	} else {
		tmp = x - (1.0 / (0.5 * t));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((Math.exp(z) * y) + (1.0 - y)) <= 2.0) {
		tmp = x - ((y * Math.expm1(z)) / t);
	} else {
		tmp = x - (1.0 / (0.5 * t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((math.exp(z) * y) + (1.0 - y)) <= 2.0:
		tmp = x - ((y * math.expm1(z)) / t)
	else:
		tmp = x - (1.0 / (0.5 * t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(exp(z) * y) + Float64(1.0 - y)) <= 2.0)
		tmp = Float64(x - Float64(Float64(y * expm1(z)) / t));
	else
		tmp = Float64(x - Float64(1.0 / Float64(0.5 * t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(N[Exp[z], $MachinePrecision] * y), $MachinePrecision] + N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 2.0], N[(x - N[(N[(y * N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(1.0 / N[(0.5 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 52.1%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 93.3

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

   5. Applied rewrites 93.3%

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

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

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

      \[\leadsto x - \frac{\log \color{blue}{\left(1 + y \cdot z\right)}}{t} \]
   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 30.2

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

   5. Applied rewrites 30.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 30.2

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

   7. Applied rewrites 30.2%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 44.5

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

   10. Applied rewrites 44.5%

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

   11. Taylor expanded in y around inf

       \[\leadsto x - \frac{1}{\frac{1}{2} \cdot \color{blue}{t}} \]
   12. Step-by-step derivation

   13. Applied rewrites 45.8%

       \[\leadsto x - \frac{1}{0.5 \cdot \color{blue}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \cdot y + \left(1 - y\right) \leq 2:\\ \;\;\;\;x - \frac{y \cdot \mathsf{expm1}\left(z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{0.5 \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \cdot y + \left(1 - y\right) \leq 2:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{0.5 \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ (* (exp z) y) (- 1.0 y)) 2.0)
   (- x (* (/ (expm1 z) t) y))
   (- x (/ 1.0 (* 0.5 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((exp(z) * y) + (1.0 - y)) <= 2.0) {
		tmp = x - ((expm1(z) / t) * y);
	} else {
		tmp = x - (1.0 / (0.5 * t));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((Math.exp(z) * y) + (1.0 - y)) <= 2.0) {
		tmp = x - ((Math.expm1(z) / t) * y);
	} else {
		tmp = x - (1.0 / (0.5 * t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((math.exp(z) * y) + (1.0 - y)) <= 2.0:
		tmp = x - ((math.expm1(z) / t) * y)
	else:
		tmp = x - (1.0 / (0.5 * t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(exp(z) * y) + Float64(1.0 - y)) <= 2.0)
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	else
		tmp = Float64(x - Float64(1.0 / Float64(0.5 * t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(N[Exp[z], $MachinePrecision] * y), $MachinePrecision] + N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 2.0], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x - N[(1.0 / N[(0.5 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 52.1%

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

   3. Taylor expanded in y around 0

      \[\leadsto x - \color{blue}{\frac{y \cdot \left(e^{z} - 1\right)}{t}} \]
   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 92.6

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

   5. Applied rewrites 92.6%

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

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

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

      \[\leadsto x - \frac{\log \color{blue}{\left(1 + y \cdot z\right)}}{t} \]
   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 30.2

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

   5. Applied rewrites 30.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 30.2

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

   7. Applied rewrites 30.2%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 44.5

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

   10. Applied rewrites 44.5%

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

   11. Taylor expanded in y around inf

       \[\leadsto x - \frac{1}{\frac{1}{2} \cdot \color{blue}{t}} \]
   12. Step-by-step derivation

   13. Applied rewrites 45.8%

       \[\leadsto x - \frac{1}{0.5 \cdot \color{blue}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \cdot y + \left(1 - y\right) \leq 2:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{0.5 \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{-1}{t}, \mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right), x\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma (/ -1.0 t) (log1p (* y (expm1 z))) x))

C

double code(double x, double y, double z, double t) {
	return fma((-1.0 / t), log1p((y * expm1(z))), x);
}

Julia

function code(x, y, z, t)
	return fma(Float64(-1.0 / t), log1p(Float64(y * expm1(z))), x)
end

Wolfram

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

Derivation

1. Initial program 57.6%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. distribute-neg-frac2 N/A

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

   6. div-inv N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 81.5%

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

   1. lift-fma.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. neg-mul-1 N/A

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

   4. distribute-rgt-in N/A

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

   5. metadata-eval N/A

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

   6. sub-neg N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-expm1.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{t}, \mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(z\right)}\right), x\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{t}, \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(z\right) \cdot y}\right), x\right) \]

   10. lift-*.f64 99.5

       \[\leadsto \mathsf{fma}\left(\frac{-1}{t}, \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(z\right) \cdot y}\right), x\right) \]

6. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{t}, \mathsf{log1p}\left(\mathsf{expm1}\left(z\right) \cdot y\right), x\right)} \]

7. Final simplification 99.5%

   \[\leadsto \mathsf{fma}\left(\frac{-1}{t}, \mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right), x\right) \]
8. Add Preprocessing

Alternative 8: 78.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x - \frac{1}{0.5 \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z, 0.5\right) \cdot y, z, y\right) \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (exp z) 0.0)
   (- x (/ 1.0 (* 0.5 t)))
   (- x (/ (* (fma (* (fma 0.16666666666666666 z 0.5) y) z y) z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x - (1.0 / (0.5 * t));
	} else {
		tmp = x - ((fma((fma(0.16666666666666666, z, 0.5) * y), z, y) * z) / t);
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x - N[(1.0 / N[(0.5 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(N[(N[(0.16666666666666666 * z + 0.5), $MachinePrecision] * y), $MachinePrecision] * z + y), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 77.7%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 41.4

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

   5. Applied rewrites 41.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 41.4

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

   7. Applied rewrites 41.4%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 76.8

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

   10. Applied rewrites 76.8%

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

   11. Taylor expanded in y around inf

       \[\leadsto x - \frac{1}{\frac{1}{2} \cdot \color{blue}{t}} \]
   12. Step-by-step derivation

   13. Applied rewrites 50.7%

       \[\leadsto x - \frac{1}{0.5 \cdot \color{blue}{t}} \]

   if 0.0 < (exp.f64 z)

   1. Initial program 49.7%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 71.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 90.9%

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

Alternative 9: 78.6% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ (* (exp z) y) (- 1.0 y)) 2.0)
   (- x (/ (* y z) t))
   (- x (/ 1.0 (* 0.5 t)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(N[Exp[z], $MachinePrecision] * y), $MachinePrecision] + N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 2.0], N[(x - N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(1.0 / N[(0.5 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 52.1%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 83.8

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

   5. Applied rewrites 83.8%

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

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

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

      \[\leadsto x - \frac{\log \color{blue}{\left(1 + y \cdot z\right)}}{t} \]
   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 30.2

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

   5. Applied rewrites 30.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 30.2

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

   7. Applied rewrites 30.2%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 44.5

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

   10. Applied rewrites 44.5%

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

   11. Taylor expanded in y around inf

       \[\leadsto x - \frac{1}{\frac{1}{2} \cdot \color{blue}{t}} \]
   12. Step-by-step derivation

   13. Applied rewrites 45.8%

       \[\leadsto x - \frac{1}{0.5 \cdot \color{blue}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \cdot y + \left(1 - y\right) \leq 2:\\ \;\;\;\;x - \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{0.5 \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 88.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{\log \left(\mathsf{fma}\left(z, y, 1\right)\right)}{t}\\ \mathbf{if}\;y \leq -2.45 \cdot 10^{+208}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+163}:\\ \;\;\;\;x - \frac{y \cdot \mathsf{expm1}\left(z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ (log (fma z y 1.0)) t))))
   (if (<= y -2.45e+208)
     t_1
     (if (<= y 1.35e+163) (- x (/ (* y (expm1 z)) t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (log(fma(z, y, 1.0)) / t);
	double tmp;
	if (y <= -2.45e+208) {
		tmp = t_1;
	} else if (y <= 1.35e+163) {
		tmp = x - ((y * expm1(z)) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(log(fma(z, y, 1.0)) / t))
	tmp = 0.0
	if (y <= -2.45e+208)
		tmp = t_1;
	elseif (y <= 1.35e+163)
		tmp = Float64(x - Float64(Float64(y * expm1(z)) / t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(N[Log[N[(z * y + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.45e+208], t$95$1, If[LessEqual[y, 1.35e+163], N[(x - N[(N[(y * N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.4499999999999998e208 or 1.35e163 < y

   1. Initial program 39.3%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 69.5

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

   5. Applied rewrites 69.5%

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

   if -2.4499999999999998e208 < y < 1.35e163

   1. Initial program 60.6%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 93.9

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

   5. Applied rewrites 93.9%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.45 \cdot 10^{+208}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(z, y, 1\right)\right)}{t}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+163}:\\ \;\;\;\;x - \frac{y \cdot \mathsf{expm1}\left(z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(z, y, 1\right)\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.7% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in z around 0

   \[\leadsto x - \frac{\log \color{blue}{\left(1 + y \cdot z\right)}}{t} \]
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 69.8

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

5. Applied rewrites 69.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 69.7

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

7. Applied rewrites 69.7%

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

8. Taylor expanded in y around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-expm1.f64 86.4

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

10. Applied rewrites 86.4%

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

11. Taylor expanded in z around 0

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

13. Applied rewrites 81.5%

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

14. Final simplification 81.5%

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

Alternative 12: 78.6% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -20000:\\ \;\;\;\;x - \frac{1}{0.5 \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(0.5, y \cdot z, y\right) \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -20000.0)
   (- x (/ 1.0 (* 0.5 t)))
   (- x (/ (* (fma 0.5 (* y z) y) z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -20000.0) {
		tmp = x - (1.0 / (0.5 * t));
	} else {
		tmp = x - ((fma(0.5, (y * z), y) * z) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -20000.0)
		tmp = Float64(x - Float64(1.0 / Float64(0.5 * t)));
	else
		tmp = Float64(x - Float64(Float64(fma(0.5, Float64(y * z), y) * z) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -20000.0], N[(x - N[(1.0 / N[(0.5 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(0.5 * N[(y * z), $MachinePrecision] + y), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2e4

   1. Initial program 77.7%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 41.4

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

   5. Applied rewrites 41.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 41.4

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

   7. Applied rewrites 41.4%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 76.8

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

   10. Applied rewrites 76.8%

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

   11. Taylor expanded in y around inf

       \[\leadsto x - \frac{1}{\frac{1}{2} \cdot \color{blue}{t}} \]
   12. Step-by-step derivation

   13. Applied rewrites 50.7%

       \[\leadsto x - \frac{1}{0.5 \cdot \color{blue}{t}} \]

   if -2e4 < z

   1. Initial program 49.7%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 91.4

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

   5. Applied rewrites 91.4%

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

   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 90.7%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -20000:\\ \;\;\;\;x - \frac{1}{0.5 \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(0.5, y \cdot z, y\right) \cdot z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.4% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -52000:\\ \;\;\;\;x - \frac{1}{0.5 \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -52000.0) (- x (/ 1.0 (* 0.5 t))) (- x (* (/ z t) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -52000.0) {
		tmp = x - (1.0 / (0.5 * t));
	} 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 <= (-52000.0d0)) then
        tmp = x - (1.0d0 / (0.5d0 * t))
    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 <= -52000.0) {
		tmp = x - (1.0 / (0.5 * t));
	} else {
		tmp = x - ((z / t) * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -52000.0:
		tmp = x - (1.0 / (0.5 * t))
	else:
		tmp = x - ((z / t) * y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -52000.0)
		tmp = Float64(x - Float64(1.0 / Float64(0.5 * t)));
	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 <= -52000.0)
		tmp = x - (1.0 / (0.5 * t));
	else
		tmp = x - ((z / t) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -52000.0], N[(x - N[(1.0 / N[(0.5 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -52000

   1. Initial program 77.7%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 41.4

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

   5. Applied rewrites 41.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 41.4

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

   7. Applied rewrites 41.4%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 76.8

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

   10. Applied rewrites 76.8%

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

   11. Taylor expanded in y around inf

       \[\leadsto x - \frac{1}{\frac{1}{2} \cdot \color{blue}{t}} \]
   12. Step-by-step derivation

   13. Applied rewrites 50.7%

       \[\leadsto x - \frac{1}{0.5 \cdot \color{blue}{t}} \]

   if -52000 < z

   1. Initial program 49.7%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 87.5

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

   5. Applied rewrites 87.5%

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

   7. Applied rewrites 89.8%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -52000:\\ \;\;\;\;x - \frac{1}{0.5 \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{t} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 75.0% 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 57.6%

   \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
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 73.9

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

5. Applied rewrites 73.9%

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

7. Applied rewrites 75.1%

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

8. Final simplification 75.1%

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

Developer Target 1: 75.5% 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 2024250 
(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)))