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

Percentage Accurate: 61.6% → 96.5%

Time: 21.7s

Alternatives: 11

Speedup: 11.3×

• could not determine a ground truth

  1. x = -1.641908445757907e-108
  2. y = 5.75931930458266e-220
  3. z = 3.734675522431704e+27
  4. t = 1.4912339223763734e-47

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: 61.6% 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: 96.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      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. lower-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. Applied rewrites 91.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 91.1%

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

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

   1. Initial program 78.6%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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. lower-*.f64 N/A

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

      2. lower-expm1.f64 97.8

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

   5. Applied rewrites 97.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 97.7

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

   7. Applied rewrites 97.7%

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

   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. lower-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. lower-*.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. lower-/.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. lower-expm1.f64 99.7

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

   10. Applied rewrites 99.7%

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

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

   1. Initial program 86.3%

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

   3. Taylor expanded in z around 0

      \[\leadsto x - \frac{\log \color{blue}{\left(1 + z \cdot \left(y + \frac{1}{2} \cdot \left(y \cdot z\right)\right)\right)}}{t} \]
   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. lower-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. lower-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. lower-*.f64 11.3

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

   5. Applied rewrites 11.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. lift-/.f64 N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 11.3

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

   7. Applied rewrites 11.3%

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

   8. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-expm1.f64 88.6

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

   10. Applied rewrites 88.6%

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

4. Final simplification 96.7%

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

Alternative 2: 91.9% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ (- 1.0 y) (* y (exp z))) 0.0)
   (fma (/ (log1p (* y z)) (* x t)) (- x) 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((log1p((y * z)) / (x * t)), -x, 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(log1p(Float64(y * z)) / Float64(x * t)), Float64(-x), 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[(N[Log[1 + N[(y * z), $MachinePrecision]], $MachinePrecision] / N[(x * t), $MachinePrecision]), $MachinePrecision] * (-x) + 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 2.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      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. lower-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. Applied rewrites 91.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 91.1%

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

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

   1. Initial program 79.6%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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. lower-*.f64 N/A

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

      2. lower-expm1.f64 88.2

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

   5. Applied rewrites 88.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 88.1

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

   7. Applied rewrites 88.1%

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

   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. lower-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. lower-*.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. lower-/.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. lower-expm1.f64 92.4

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

   10. Applied rewrites 92.4%

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

4. Final simplification 92.1%

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

Alternative 3: 89.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((1.0 - y) + (y * exp(z))) <= 0.0) {
		tmp = fma((log1p((y * z)) / (x * t)), -x, x);
	} else {
		tmp = x - (y * (expm1(z) / t));
	}
	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(log1p(Float64(y * z)) / Float64(x * t)), Float64(-x), x);
	else
		tmp = Float64(x - Float64(y * Float64(expm1(z) / t)));
	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[(N[Log[1 + N[(y * z), $MachinePrecision]], $MachinePrecision] / N[(x * t), $MachinePrecision]), $MachinePrecision] * (-x) + x), $MachinePrecision], N[(x - N[(y * N[(N[(Exp[z] - 1), $MachinePrecision] / 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))) < 0.0

   1. Initial program 2.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      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. lower-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. Applied rewrites 91.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 91.1%

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

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

   1. Initial program 79.6%

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

   3. Taylor expanded in y around 0

      \[\leadsto x - \color{blue}{\frac{y \cdot \left(e^{z} - 1\right)}{t}} \]
   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. lower-*.f64 N/A

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

      4. div-sub N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 89.5

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

   5. Applied rewrites 89.5%

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

4. Final simplification 89.9%

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

Alternative 4: 87.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{+228}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(y, z, 1\right)\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 5.2e+228)
   (- x (* y (/ (expm1 z) t)))
   (- x (/ (log (fma y z 1.0)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 5.2e+228) {
		tmp = x - (y * (expm1(z) / t));
	} else {
		tmp = x - (log(fma(y, z, 1.0)) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 5.2e+228)
		tmp = Float64(x - Float64(y * Float64(expm1(z) / t)));
	else
		tmp = Float64(x - Float64(log(fma(y, z, 1.0)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 5.2e+228], N[(x - N[(y * N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(y * z + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.20000000000000015e228

   1. Initial program 64.7%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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. lower-*.f64 N/A

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

      4. div-sub N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 88.3

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

   5. Applied rewrites 88.3%

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

   if 5.20000000000000015e228 < y

   1. Initial program 0.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 86.1

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

   5. Applied rewrites 86.1%

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

Alternative 5: 82.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+18}:\\ \;\;\;\;x - \frac{\log 1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \left(z \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(0.041666666666666664, \frac{z}{t}, \frac{0.16666666666666666}{t}\right), \frac{0.5}{t}\right), \frac{1}{t}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.9e+18)
   (- x (/ (log 1.0) t))
   (-
    x
    (*
     y
     (*
      z
      (fma
       z
       (fma
        z
        (fma 0.041666666666666664 (/ z t) (/ 0.16666666666666666 t))
        (/ 0.5 t))
       (/ 1.0 t)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.9e+18) {
		tmp = x - (log(1.0) / t);
	} else {
		tmp = x - (y * (z * fma(z, fma(z, fma(0.041666666666666664, (z / t), (0.16666666666666666 / t)), (0.5 / t)), (1.0 / t))));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.9e+18)
		tmp = Float64(x - Float64(log(1.0) / t));
	else
		tmp = Float64(x - Float64(y * Float64(z * fma(z, fma(z, fma(0.041666666666666664, Float64(z / t), Float64(0.16666666666666666 / t)), Float64(0.5 / t)), Float64(1.0 / t)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.9e+18], N[(x - N[(N[Log[1.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(y * N[(z * N[(z * N[(z * N[(0.041666666666666664 * N[(z / t), $MachinePrecision] + N[(0.16666666666666666 / t), $MachinePrecision]), $MachinePrecision] + N[(0.5 / t), $MachinePrecision]), $MachinePrecision] + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.9e18

   1. Initial program 75.6%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 56.0%

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

   if -2.9e18 < z

   1. Initial program 58.1%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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. lower-*.f64 N/A

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

      4. div-sub N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 89.4

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

   5. Applied rewrites 89.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 90.0%

      \[\leadsto x - y \cdot \left(z \cdot \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.041666666666666664}{t}, \frac{0.16666666666666666}{t}\right), \frac{0.5}{t}\right), \frac{1}{t}\right)}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 90.0%

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

Alternative 6: 86.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x, double y, double z, double t) {
	return x - (y * (Math.expm1(z) / t));
}

Python

def code(x, y, z, t):
	return x - (y * (math.expm1(z) / t))

Julia

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

Wolfram

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

Derivation

1. Initial program 62.9%

   \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
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. lower-*.f64 N/A

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

   4. div-sub N/A

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

   5. lower-/.f64 N/A

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

   6. lower-expm1.f64 86.1

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

5. Applied rewrites 86.1%

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

Alternative 7: 82.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \left(z \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(0.041666666666666664, \frac{z}{t}, \frac{0.16666666666666666}{t}\right), \frac{0.5}{t}\right), \frac{1}{t}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.9e+18)
   (/ 1.0 (/ 1.0 x))
   (-
    x
    (*
     y
     (*
      z
      (fma
       z
       (fma
        z
        (fma 0.041666666666666664 (/ z t) (/ 0.16666666666666666 t))
        (/ 0.5 t))
       (/ 1.0 t)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.9e+18) {
		tmp = 1.0 / (1.0 / x);
	} else {
		tmp = x - (y * (z * fma(z, fma(z, fma(0.041666666666666664, (z / t), (0.16666666666666666 / t)), (0.5 / t)), (1.0 / t))));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.9e+18)
		tmp = Float64(1.0 / Float64(1.0 / x));
	else
		tmp = Float64(x - Float64(y * Float64(z * fma(z, fma(z, fma(0.041666666666666664, Float64(z / t), Float64(0.16666666666666666 / t)), Float64(0.5 / t)), Float64(1.0 / t)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.9e+18], N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x - N[(y * N[(z * N[(z * N[(z * N[(0.041666666666666664 * N[(z / t), $MachinePrecision] + N[(0.16666666666666666 / t), $MachinePrecision]), $MachinePrecision] + N[(0.5 / t), $MachinePrecision]), $MachinePrecision] + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.9e18

   1. Initial program 75.6%

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

   3. Taylor expanded in z around 0

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

      1. lower-*.f64 31.9

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

   5. Applied rewrites 31.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 15.8%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 55.9

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   10. Applied rewrites 55.9%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   if -2.9e18 < z

   1. Initial program 58.1%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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. lower-*.f64 N/A

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

      4. div-sub N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 89.4

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

   5. Applied rewrites 89.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 90.0%

      \[\leadsto x - y \cdot \left(z \cdot \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.041666666666666664}{t}, \frac{0.16666666666666666}{t}\right), \frac{0.5}{t}\right), \frac{1}{t}\right)}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 90.0%

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

Alternative 8: 82.2% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \left(z \cdot \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right)}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.9e+18)
   (/ 1.0 (/ 1.0 x))
   (-
    x
    (*
     y
     (*
      z
      (/
       (fma z (fma z (fma z 0.041666666666666664 0.16666666666666666) 0.5) 1.0)
       t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.9e+18) {
		tmp = 1.0 / (1.0 / x);
	} else {
		tmp = x - (y * (z * (fma(z, fma(z, fma(z, 0.041666666666666664, 0.16666666666666666), 0.5), 1.0) / t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.9e+18)
		tmp = Float64(1.0 / Float64(1.0 / x));
	else
		tmp = Float64(x - Float64(y * Float64(z * Float64(fma(z, fma(z, fma(z, 0.041666666666666664, 0.16666666666666666), 0.5), 1.0) / t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.9e+18], N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x - N[(y * N[(z * N[(N[(z * N[(z * N[(z * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.9e18

   1. Initial program 75.6%

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

   3. Taylor expanded in z around 0

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

      1. lower-*.f64 31.9

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

   5. Applied rewrites 31.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 15.8%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 55.9

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   10. Applied rewrites 55.9%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   if -2.9e18 < z

   1. Initial program 58.1%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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. lower-*.f64 N/A

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

      4. div-sub N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 89.4

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

   5. Applied rewrites 89.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 90.0%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 90.0%

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

Alternative 9: 82.2% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z \cdot \mathsf{fma}\left(z, -0.5, -1\right)}{-t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.9e+18)
   (/ 1.0 (/ 1.0 x))
   (- x (* y (/ (* z (fma z -0.5 -1.0)) (- t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.9e+18) {
		tmp = 1.0 / (1.0 / x);
	} else {
		tmp = x - (y * ((z * fma(z, -0.5, -1.0)) / -t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.9e+18)
		tmp = Float64(1.0 / Float64(1.0 / x));
	else
		tmp = Float64(x - Float64(y * Float64(Float64(z * fma(z, -0.5, -1.0)) / Float64(-t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.9e+18], N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x - N[(y * N[(N[(z * N[(z * -0.5 + -1.0), $MachinePrecision]), $MachinePrecision] / (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.9e18

   1. Initial program 75.6%

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

   3. Taylor expanded in z around 0

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

      1. lower-*.f64 31.9

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

   5. Applied rewrites 31.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 15.8%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 55.9

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   10. Applied rewrites 55.9%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   if -2.9e18 < z

   1. Initial program 58.1%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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. lower-*.f64 N/A

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

      4. div-sub N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 89.4

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

   5. Applied rewrites 89.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 89.8%

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

   9. Taylor expanded in t around -inf

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

   11. Applied rewrites 89.8%

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

Alternative 10: 81.8% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+58}:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.8e+58) (/ 1.0 (/ 1.0 x)) (- x (* y (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.8e+58) {
		tmp = 1.0 / (1.0 / 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 <= (-2.8d+58)) then
        tmp = 1.0d0 / (1.0d0 / 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 <= -2.8e+58) {
		tmp = 1.0 / (1.0 / x);
	} else {
		tmp = x - (y * (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -2.8e+58:
		tmp = 1.0 / (1.0 / x)
	else:
		tmp = x - (y * (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.8e+58)
		tmp = Float64(1.0 / Float64(1.0 / x));
	else
		tmp = Float64(x - Float64(y * Float64(z / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -2.8e+58)
		tmp = 1.0 / (1.0 / x);
	else
		tmp = x - (y * (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.8e+58], N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x - N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.7999999999999998e58

   1. Initial program 76.8%

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

   3. Taylor expanded in z around 0

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

      1. lower-*.f64 29.4

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

   5. Applied rewrites 29.4%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 17.3%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 59.5

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   10. Applied rewrites 59.5%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   if -2.7999999999999998e58 < z

   1. Initial program 58.9%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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. lower-*.f64 N/A

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

      4. div-sub N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 88.2

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

   5. Applied rewrites 88.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 86.4%

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

Alternative 11: 74.5% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.9%

   \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
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. lower-*.f64 N/A

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

   4. div-sub N/A

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

   5. lower-/.f64 N/A

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

   6. lower-expm1.f64 86.1

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

5. Applied rewrites 86.1%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 73.1%

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

Developer Target 1: 74.3% 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 2024221 
(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)))