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

Percentage Accurate: 61.9% → 95.6%

Time: 19.8s

Alternatives: 9

Speedup: 11.3×

• could not determine a ground truth

  1. x = 2.205565165809881e-208
  2. y = 4.952840261925292e-247
  3. z = 1.0593005620021495e+294
  4. t = 1.9058593127850148e-160

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.6% accurate, 0.5× speedup

Math

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

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 1.0)
       (- x (* y (/ (expm1 z) t)))
       (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 <= 1.0) {
		tmp = x - (y * (expm1(z) / t));
	} 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 <= 1.0)
		tmp = Float64(x - Float64(y * Float64(expm1(z) / t)));
	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, 1.0], N[(x - N[(y * N[(N[(Exp[z] - 1), $MachinePrecision] / t), $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 1.8%

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

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      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 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. 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 89.6%

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

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

   1. Initial program 84.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-expm1.f64 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-expm1.f64 99.9

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

   8. Applied rewrites 99.9%

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

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

   1. Initial program 96.4%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-expm1.f64 96.4

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

   5. Applied rewrites 96.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 96.4

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

   7. Applied rewrites 96.4%

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

4. Final simplification 97.0%

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

Alternative 2: 95.6% accurate, 0.5× speedup

Math

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

FPCore

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

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 <= 1.0) {
		tmp = x - (y * (expm1(z) / t));
	} else {
		tmp = x - (log((y * expm1(z))) / t);
	}
	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 <= 1.0)
		tmp = Float64(x - Float64(y * Float64(expm1(z) / t)));
	else
		tmp = Float64(x - Float64(log(Float64(y * expm1(z))) / t));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 1.8%

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

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      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 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. 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 89.6%

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

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

   1. Initial program 84.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-expm1.f64 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-expm1.f64 99.9

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

   8. Applied rewrites 99.9%

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

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

   1. Initial program 96.4%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-expm1.f64 96.4

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

   5. Applied rewrites 96.4%

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

4. Final simplification 97.0%

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

Alternative 3: 90.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1 - y\right) + y \cdot e^{z}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{log1p}\left(y \cdot z\right)}{x \cdot t}, -x, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+75}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log 1}{t}\\ \end{array} \end{array} \]

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 5e+75) (- x (* y (/ (expm1 z) t))) (- x (/ (log 1.0) t))))))

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 <= 5e+75) {
		tmp = x - (y * (expm1(z) / t));
	} else {
		tmp = x - (log(1.0) / t);
	}
	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 <= 5e+75)
		tmp = Float64(x - Float64(y * Float64(expm1(z) / t)));
	else
		tmp = Float64(x - Float64(log(1.0) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(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, 5e+75], N[(x - N[(y * N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[1.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 1.8%

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

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      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 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. 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 89.6%

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

   if 0.0 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))) < 5.0000000000000002e75

   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 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 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-expm1.f64 97.7

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

   8. Applied rewrites 97.7%

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

   if 5.0000000000000002e75 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))

   1. Initial program 93.4%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 56.3%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - y\right) + y \cdot e^{z} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{log1p}\left(y \cdot z\right)}{x \cdot t}, -x, x\right)\\ \mathbf{elif}\;\left(1 - y\right) + y \cdot e^{z} \leq 5 \cdot 10^{+75}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log 1}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{\log \left(\mathsf{fma}\left(y, z, 1\right)\right)}{t}\\ \mathbf{if}\;y \leq -1.15 \cdot 10^{+155}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+179}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ (log (fma y z 1.0)) t))))
   (if (<= y -1.15e+155)
     t_1
     (if (<= y 2.2e+179) (- x (* y (/ (expm1 z) t))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (log(fma(y, z, 1.0)) / t);
	double tmp;
	if (y <= -1.15e+155) {
		tmp = t_1;
	} else if (y <= 2.2e+179) {
		tmp = x - (y * (expm1(z) / t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(log(fma(y, z, 1.0)) / t))
	tmp = 0.0
	if (y <= -1.15e+155)
		tmp = t_1;
	elseif (y <= 2.2e+179)
		tmp = Float64(x - Float64(y * Float64(expm1(z) / t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(N[Log[N[(y * z + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.15e+155], t$95$1, If[LessEqual[y, 2.2e+179], N[(x - N[(y * N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.14999999999999999e155 or 2.2e179 < y

   1. Initial program 34.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 78.8

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

   5. Applied rewrites 78.8%

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

   if -1.14999999999999999e155 < y < 2.2e179

   1. Initial program 69.8%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 94.7

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

   5. Applied rewrites 94.7%

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

   6. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-expm1.f64 95.6

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

   8. Applied rewrites 95.6%

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

Alternative 5: 87.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+156}:\\ \;\;\;\;x - \frac{\log 1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -4.5e+156) (- x (/ (log 1.0) t)) (- x (* y (/ (expm1 z) t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -4.5e+156) {
		tmp = x - (log(1.0) / t);
	} else {
		tmp = x - (y * (expm1(z) / t));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -4.5e+156) {
		tmp = x - (Math.log(1.0) / t);
	} else {
		tmp = x - (y * (Math.expm1(z) / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -4.5e+156:
		tmp = x - (math.log(1.0) / t)
	else:
		tmp = x - (y * (math.expm1(z) / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -4.5e+156)
		tmp = Float64(x - Float64(log(1.0) / t));
	else
		tmp = Float64(x - Float64(y * Float64(expm1(z) / t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -4.5e+156], N[(x - N[(N[Log[1.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(y * N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.50000000000000031e156

   1. Initial program 46.5%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 66.6%

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

   if -4.50000000000000031e156 < y

   1. Initial program 66.6%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 92.1

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

   5. Applied rewrites 92.1%

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

   6. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-expm1.f64 93.2

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

   8. Applied rewrites 93.2%

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

Alternative 6: 82.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -680000000000:\\ \;\;\;\;x - \frac{\log 1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{fma}\left(z, z \cdot \mathsf{fma}\left(z, 0.16666666666666666, 0.5\right), z\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -680000000000.0)
   (- x (/ (log 1.0) t))
   (- x (* y (/ (fma z (* z (fma z 0.16666666666666666 0.5)) z) t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -680000000000.0) {
		tmp = x - (log(1.0) / t);
	} else {
		tmp = x - (y * (fma(z, (z * fma(z, 0.16666666666666666, 0.5)), z) / t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -680000000000.0)
		tmp = Float64(x - Float64(log(1.0) / t));
	else
		tmp = Float64(x - Float64(y * Float64(fma(z, Float64(z * fma(z, 0.16666666666666666, 0.5)), z) / t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -680000000000.0], N[(x - N[(N[Log[1.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(y * N[(N[(z * N[(z * N[(z * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.8e11

   1. Initial program 77.5%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 66.5%

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

   if -6.8e11 < z

   1. Initial program 60.2%

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

   3. Taylor expanded in y around 0

      \[\leadsto x - \frac{\color{blue}{y \cdot \left(e^{z} - 1\right)}}{t} \]
   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 89.2

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

   5. Applied rewrites 89.2%

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

   6. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-expm1.f64 90.6

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

   8. Applied rewrites 90.6%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 90.3%

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

Alternative 7: 74.6% 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 64.8%

   \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
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 87.2

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

5. Applied rewrites 87.2%

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

6. Taylor expanded in y around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-expm1.f64 88.2

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

8. Applied rewrites 88.2%

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

9. Taylor expanded in z around 0

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

11. Applied rewrites 75.7%

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

Alternative 8: 72.5% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 64.8%

   \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
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 87.2

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

5. Applied rewrites 87.2%

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

6. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. associate-*l/ N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 74.2

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

8. Applied rewrites 74.2%

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

Alternative 9: 13.1% accurate, 11.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.8%

   \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
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 87.2

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

5. Applied rewrites 87.2%

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

6. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. associate-*l/ N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 74.2

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

8. Applied rewrites 74.2%

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

9. Taylor expanded in y around inf

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

11. Applied rewrites 12.1%

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

12. Final simplification 12.1%

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

Developer Target 1: 74.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024220 
(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)))