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

Percentage Accurate: 60.8% → 98.3%

Time: 21.1s

Alternatives: 11

Speedup: 226.0×

• could not determine a ground truth

  1. x = -1.3290281807339983e-41
  2. y = 4.9833443045976544e-8
  3. z = 1.1238225813387475e+276
  4. t = 4.308304812786354e-270

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: 60.8% 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: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

5. Simplified 89.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t \cdot x}, -x, x\right)} \]
6. Step-by-step derivation

   1. div-inv N/A

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

   2. associate-*l* N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

7. Applied egg-rr 89.4%

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

8. Taylor expanded in t around 0

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

   1. /-lowering-/.f64 99.2

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

10. Simplified 99.2%

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

Alternative 2: 94.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 2.2%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. sub-neg N/A

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

      5. associate-+l+ N/A

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

      6. accelerator-lowering-log1p.f64 N/A

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. exp-lowering-exp.f64 N/A

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

      10. neg-lowering-neg.f64 67.5

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

   4. Applied egg-rr 67.5%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

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

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

      5. accelerator-lowering-fma.f64 N/A

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in z around 0

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

      1. *-lowering-*.f64 99.8

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

   9. Simplified 99.8%

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

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

   1. Initial program 83.5%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. sub-neg N/A

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

      5. associate-+l+ N/A

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

      6. accelerator-lowering-log1p.f64 N/A

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. exp-lowering-exp.f64 N/A

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

      10. neg-lowering-neg.f64 91.6

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

   4. Applied egg-rr 91.6%

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

   5. Taylor expanded in y around 0

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

      1. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. accelerator-lowering-expm1.f64 92.8

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

   7. Simplified 92.8%

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

4. Final simplification 94.5%

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

Alternative 3: 90.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.1 \cdot 10^{+33}:\\ \;\;\;\;x - \frac{y \cdot \mathsf{expm1}\left(z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{t}, -\mathsf{log1p}\left(y \cdot z\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -5.1e+33)
   (- x (/ (* y (expm1 z)) t))
   (fma (/ 1.0 t) (- (log1p (* y z))) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -5.1e+33) {
		tmp = x - ((y * expm1(z)) / t);
	} else {
		tmp = fma((1.0 / t), -log1p((y * z)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -5.1e+33)
		tmp = Float64(x - Float64(Float64(y * expm1(z)) / t));
	else
		tmp = fma(Float64(1.0 / t), Float64(-log1p(Float64(y * z))), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -5.1e+33], N[(x - N[(N[(y * N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t), $MachinePrecision] * (-N[Log[1 + N[(y * z), $MachinePrecision]], $MachinePrecision]) + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.0999999999999999e33

   1. Initial program 84.5%

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

   3. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 N/A

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

      2. accelerator-lowering-expm1.f64 81.8

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

   5. Simplified 81.8%

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

   if -5.0999999999999999e33 < z

   1. Initial program 55.6%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. sub-neg N/A

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

      5. associate-+l+ N/A

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

      6. accelerator-lowering-log1p.f64 N/A

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. exp-lowering-exp.f64 N/A

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

      10. neg-lowering-neg.f64 80.0

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

   4. Applied egg-rr 80.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

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

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

      5. accelerator-lowering-fma.f64 N/A

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in z around 0

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

      1. *-lowering-*.f64 97.3

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

   9. Simplified 97.3%

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

Alternative 4: 90.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{+33}:\\ \;\;\;\;x - \frac{y \cdot \mathsf{expm1}\left(z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -4.9e+33) (- x (/ (* y (expm1 z)) t)) (- x (/ (log1p (* y z)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4.9e+33) {
		tmp = x - ((y * expm1(z)) / t);
	} else {
		tmp = x - (log1p((y * z)) / t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4.9e+33) {
		tmp = x - ((y * Math.expm1(z)) / t);
	} else {
		tmp = x - (Math.log1p((y * z)) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -4.9e+33:
		tmp = x - ((y * math.expm1(z)) / t)
	else:
		tmp = x - (math.log1p((y * z)) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -4.9e+33)
		tmp = Float64(x - Float64(Float64(y * expm1(z)) / t));
	else
		tmp = Float64(x - Float64(log1p(Float64(y * z)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -4.9e+33], N[(x - N[(N[(y * N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[1 + N[(y * z), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.90000000000000014e33

   1. Initial program 84.5%

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

   3. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 N/A

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

      2. accelerator-lowering-expm1.f64 81.8

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

   5. Simplified 81.8%

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

   if -4.90000000000000014e33 < z

   1. Initial program 55.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 90.6%

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

   6. Taylor expanded in z around 0

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

      1. *-lowering-*.f64 90.3

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

   8. Simplified 90.3%

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

   9. Taylor expanded in t around inf

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. --lowering--.f64 N/A

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

       4. /-lowering-/.f64 N/A

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

       5. accelerator-lowering-log1p.f64 N/A

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

       6. *-lowering-*.f64 97.3

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

   11. Simplified 97.3%

       \[\leadsto \color{blue}{x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 87.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.12 \cdot 10^{+52}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.12e+52) x (- x (/ (log1p (* y z)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.12e+52) {
		tmp = x;
	} else {
		tmp = x - (log1p((y * z)) / t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.12e+52) {
		tmp = x;
	} else {
		tmp = x - (Math.log1p((y * z)) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.12e+52:
		tmp = x
	else:
		tmp = x - (math.log1p((y * z)) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.12e+52)
		tmp = x;
	else
		tmp = Float64(x - Float64(log1p(Float64(y * z)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.12e+52], x, N[(x - N[(N[Log[1 + N[(y * z), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.12000000000000002e52

   1. Initial program 84.5%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 67.9%

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

   if -1.12000000000000002e52 < z

   1. Initial program 56.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 90.9%

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

   6. Taylor expanded in z around 0

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

      1. *-lowering-*.f64 90.1

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

   8. Simplified 90.1%

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

   9. Taylor expanded in t around inf

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. --lowering--.f64 N/A

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

       4. /-lowering-/.f64 N/A

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

       5. accelerator-lowering-log1p.f64 N/A

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

       6. *-lowering-*.f64 96.9

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

   11. Simplified 96.9%

       \[\leadsto \color{blue}{x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 81.3% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{-59}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, z \cdot \frac{\mathsf{fma}\left(z, 0.5, 1\right)}{t}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -9.5e-59) x (fma (- y) (* z (/ (fma z 0.5 1.0) t)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -9.5e-59) {
		tmp = x;
	} else {
		tmp = fma(-y, (z * (fma(z, 0.5, 1.0) / t)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -9.5e-59)
		tmp = x;
	else
		tmp = fma(Float64(-y), Float64(z * Float64(fma(z, 0.5, 1.0) / t)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -9.5e-59], x, N[((-y) * N[(z * N[(N[(z * 0.5 + 1.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.4999999999999994e-59

   1. Initial program 85.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 73.9%

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

   if -9.4999999999999994e-59 < z

   1. Initial program 52.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 80.9

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

   5. Simplified 80.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. neg-lowering-neg.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 92.6

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

   8. Simplified 92.6%

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

Alternative 7: 71.1% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-300}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-270}:\\ \;\;\;\;y \cdot \frac{z}{-t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2.5e-300) x (if (<= t 2.8e-270) (* y (/ z (- t))) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.5e-300) {
		tmp = x;
	} else if (t <= 2.8e-270) {
		tmp = y * (z / -t);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-2.5d-300)) then
        tmp = x
    else if (t <= 2.8d-270) then
        tmp = y * (z / -t)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.5e-300) {
		tmp = x;
	} else if (t <= 2.8e-270) {
		tmp = y * (z / -t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -2.5e-300:
		tmp = x
	elif t <= 2.8e-270:
		tmp = y * (z / -t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2.5e-300)
		tmp = x;
	elseif (t <= 2.8e-270)
		tmp = Float64(y * Float64(z / Float64(-t)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -2.5e-300)
		tmp = x;
	elseif (t <= 2.8e-270)
		tmp = y * (z / -t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2.5e-300], x, If[LessEqual[t, 2.8e-270], N[(y * N[(z / (-t)), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -2.49999999999999998e-300 or 2.7999999999999999e-270 < t

   1. Initial program 67.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 78.9%

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

   if -2.49999999999999998e-300 < t < 2.7999999999999999e-270

   1. Initial program 10.8%

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

   3. Taylor expanded in z around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 48.5%

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

   6. Taylor expanded in z around 0

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

   8. Simplified 85.4%

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

   9. Taylor expanded in x around 0

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

       1. associate-*r/ N/A

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

       2. mul-1-neg N/A

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

       3. /-lowering-/.f64 N/A

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

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

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

       5. mul-1-neg N/A

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

       6. *-lowering-*.f64 N/A

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

       7. mul-1-neg N/A

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

       8. neg-lowering-neg.f64 78.3

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

   11. Simplified 78.3%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. /-lowering-/.f64 N/A

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

       5. neg-lowering-neg.f64 78.4

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

   13. Applied egg-rr 78.4%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-300}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-270}:\\ \;\;\;\;y \cdot \frac{z}{-t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.3% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{-59}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{z}{t}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -9.5e-59) x (fma (- y) (/ z t) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -9.5e-59) {
		tmp = x;
	} else {
		tmp = fma(-y, (z / t), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -9.5e-59)
		tmp = x;
	else
		tmp = fma(Float64(-y), Float64(z / t), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -9.5e-59], x, N[((-y) * N[(z / t), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.4999999999999994e-59

   1. Initial program 85.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 73.9%

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

   if -9.4999999999999994e-59 < z

   1. Initial program 52.2%

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

   3. Taylor expanded in z around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 74.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. neg-lowering-neg.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. +-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. accelerator-lowering-fma.f64 92.6

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

   8. Simplified 92.6%

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

   9. Taylor expanded in z around 0

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

       1. /-lowering-/.f64 92.4

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

   11. Simplified 92.4%

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

Alternative 9: 80.4% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{-59}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -9.5e-59) x (- x (/ (* y z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -9.5e-59) {
		tmp = x;
	} else {
		tmp = x - ((y * z) / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-9.5d-59)) then
        tmp = x
    else
        tmp = x - ((y * z) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -9.5e-59) {
		tmp = x;
	} else {
		tmp = x - ((y * z) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -9.5e-59:
		tmp = x
	else:
		tmp = x - ((y * z) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -9.5e-59)
		tmp = x;
	else
		tmp = Float64(x - Float64(Float64(y * z) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -9.5e-59)
		tmp = x;
	else
		tmp = x - ((y * z) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -9.5e-59], x, N[(x - N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.4999999999999994e-59

   1. Initial program 85.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 73.9%

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

   if -9.4999999999999994e-59 < z

   1. Initial program 52.2%

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

   3. Taylor expanded in z around 0

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

      1. *-lowering-*.f64 91.9

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

   5. Simplified 91.9%

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

Alternative 10: 78.3% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{-59}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -9.5e-59) x (- x (* z (/ y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -9.5e-59) {
		tmp = x;
	} else {
		tmp = x - (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) :: tmp
    if (z <= (-9.5d-59)) then
        tmp = x
    else
        tmp = x - (z * (y / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -9.5e-59) {
		tmp = x;
	} else {
		tmp = x - (z * (y / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -9.5e-59:
		tmp = x
	else:
		tmp = x - (z * (y / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -9.5e-59)
		tmp = x;
	else
		tmp = Float64(x - Float64(z * Float64(y / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -9.5e-59)
		tmp = x;
	else
		tmp = x - (z * (y / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -9.5e-59], x, N[(x - N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.4999999999999994e-59

   1. Initial program 85.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 73.9%

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

   if -9.4999999999999994e-59 < z

   1. Initial program 52.2%

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

   3. Taylor expanded in z around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 74.7%

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

   6. Taylor expanded in z around 0

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

   8. Simplified 91.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 87.8

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

   10. Applied egg-rr 87.8%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{-59}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.0% accurate, 226.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

3. Taylor expanded in x around inf

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

5. Simplified 75.4%

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

Developer Target 1: 74.5% 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 2024199 
(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)))