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

Percentage Accurate: 61.3% → 99.1%

Time: 19.7s

Alternatives: 8

Speedup: 226.0×

• could not determine a ground truth

  1. x = -1.8582104231742828e+83
  2. y = 3.2583876580539714e-27
  3. z = 8.334024389403187e+128
  4. t = -8.458707096902079e-236

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: 61.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1 - y\right) + y \cdot e^{z}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \mathbf{elif}\;t\_1 \leq 1.0000005:\\ \;\;\;\;\mathsf{fma}\left(y, 0 - \frac{\mathsf{expm1}\left(z\right)}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (- 1.0 y) (* y (exp z)))))
   (if (<= t_1 0.0)
     (- x (/ (log1p (* y z)) t))
     (if (<= t_1 1.0000005)
       (fma y (- 0.0 (/ (expm1 z) t)) x)
       (- 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 = x - (log1p((y * z)) / t);
	} else if (t_1 <= 1.0000005) {
		tmp = fma(y, (0.0 - (expm1(z) / t)), x);
	} 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 = Float64(x - Float64(log1p(Float64(y * z)) / t));
	elseif (t_1 <= 1.0000005)
		tmp = fma(y, Float64(0.0 - Float64(expm1(z) / t)), x);
	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[(x - N[(N[Log[1 + N[(y * z), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0000005], N[(y * N[(0.0 - N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + x), $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. Step-by-step derivation

      1. sub-neg N/A

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

      2. associate-+l+ N/A

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

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

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

      4. neg-mul-1 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-out N/A

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

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

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

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

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

      9. exp-lowering-exp.f64 59.9

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

   4. Applied egg-rr 59.9%

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

   5. Taylor expanded in z around 0

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

   7. Simplified 99.9%

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

   if 0.0 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))) < 1.0000005000000001

   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. /-lowering-/.f64 N/A

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

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

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

      3. accelerator-lowering-expm1.f64 98.1

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

   5. Simplified 98.1%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      8. accelerator-lowering-expm1.f64 99.8

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

   7. Applied egg-rr 99.8%

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

   if 1.0000005000000001 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))

   1. Initial program 88.6%

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

   3. Taylor expanded in y around inf

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

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

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

      2. accelerator-lowering-expm1.f64 89.8

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

   5. Simplified 89.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - y\right) + y \cdot e^{z} \leq 0:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \mathbf{elif}\;\left(1 - y\right) + y \cdot e^{z} \leq 1.0000005:\\ \;\;\;\;\mathsf{fma}\left(y, 0 - \frac{\mathsf{expm1}\left(z\right)}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1 - y\right) + y \cdot e^{z}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(y, 0 - \frac{\mathsf{expm1}\left(z\right)}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (- 1.0 y) (* y (exp z)))))
   (if (<= t_1 0.0)
     (- x (/ (log1p (* y z)) t))
     (if (<= t_1 5e+28) (fma y (- 0.0 (/ (expm1 z) t)) x) 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 = x - (log1p((y * z)) / t);
	} else if (t_1 <= 5e+28) {
		tmp = fma(y, (0.0 - (expm1(z) / t)), x);
	} else {
		tmp = 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 = Float64(x - Float64(log1p(Float64(y * z)) / t));
	elseif (t_1 <= 5e+28)
		tmp = fma(y, Float64(0.0 - Float64(expm1(z) / t)), x);
	else
		tmp = 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[(x - N[(N[Log[1 + N[(y * z), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+28], N[(y * N[(0.0 - N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], x]]]

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. Step-by-step derivation

      1. sub-neg N/A

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

      2. associate-+l+ N/A

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

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

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

      4. neg-mul-1 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-out N/A

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

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

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

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

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

      9. exp-lowering-exp.f64 59.9

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

   4. Applied egg-rr 59.9%

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

   5. Taylor expanded in z around 0

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

   7. Simplified 99.9%

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

   if 0.0 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))) < 4.99999999999999957e28

   1. Initial program 80.1%

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

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

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

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

      3. accelerator-lowering-expm1.f64 97.1

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

   5. Simplified 97.1%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      8. accelerator-lowering-expm1.f64 98.7

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

   7. Applied egg-rr 98.7%

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

   if 4.99999999999999957e28 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))

   1. Initial program 86.7%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 45.9%

      \[\leadsto \color{blue}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - y\right) + y \cdot e^{z} \leq 0:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \mathbf{elif}\;\left(1 - y\right) + y \cdot e^{z} \leq 5 \cdot 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(y, 0 - \frac{\mathsf{expm1}\left(z\right)}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x, double y, double z, double t) {
	return x - (Math.log1p((y * Math.expm1(z))) / t);
}

Python

def code(x, y, z, t):
	return x - (math.log1p((y * math.expm1(z))) / t)

Julia

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

Wolfram

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

Derivation

1. Initial program 61.7%

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

   1. sub-neg N/A

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

   2. associate-+l+ N/A

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

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

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

   4. neg-mul-1 N/A

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

   5. *-commutative N/A

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

   6. distribute-rgt-out N/A

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

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

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

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

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

   9. exp-lowering-exp.f64 82.7

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

4. Applied egg-rr 82.7%

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

   3. sub-neg N/A

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

   4. accelerator-lowering-expm1.f64 97.7

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

6. Applied egg-rr 97.7%

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

Alternative 4: 91.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1050000:\\ \;\;\;\;x - \frac{y \cdot \mathsf{expm1}\left(z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1050000.0)
   (- x (/ (* y (expm1 z)) t))
   (- x (/ (log1p (* y z)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1050000.0) {
		tmp = x - ((y * expm1(z)) / t);
	} else {
		tmp = x - (log1p((y * z)) / t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1050000.0) {
		tmp = x - ((y * Math.expm1(z)) / t);
	} else {
		tmp = x - (Math.log1p((y * z)) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1050000.0:
		tmp = x - ((y * math.expm1(z)) / t)
	else:
		tmp = x - (math.log1p((y * z)) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1050000.0)
		tmp = Float64(x - Float64(Float64(y * expm1(z)) / t));
	else
		tmp = Float64(x - Float64(log1p(Float64(y * z)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1050000.0], N[(x - N[(N[(y * N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[1 + N[(y * z), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.05e6

   1. Initial program 77.4%

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

   3. Taylor expanded in y around 0

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

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

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

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

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

      3. accelerator-lowering-expm1.f64 80.8

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

   5. Simplified 80.8%

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

   if -1.05e6 < z

   1. Initial program 55.5%

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

      1. sub-neg N/A

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

      2. associate-+l+ N/A

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

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

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

      4. neg-mul-1 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-out N/A

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

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

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

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

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

      9. exp-lowering-exp.f64 75.9

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

   4. Applied egg-rr 75.9%

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

   5. Taylor expanded in z around 0

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

   7. Simplified 96.5%

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

Alternative 5: 87.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+58}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -9e+58) x (- x (/ (log1p (* y z)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -9e+58) {
		tmp = x;
	} else {
		tmp = x - (log1p((y * z)) / t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -9e+58) {
		tmp = x;
	} else {
		tmp = x - (Math.log1p((y * z)) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -9e+58:
		tmp = x
	else:
		tmp = x - (math.log1p((y * z)) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -9e+58)
		tmp = x;
	else
		tmp = Float64(x - Float64(log1p(Float64(y * z)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -9e+58], x, N[(x - N[(N[Log[1 + N[(y * z), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.9999999999999996e58

   1. Initial program 76.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 63.1%

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

   if -8.9999999999999996e58 < z

   1. Initial program 56.7%

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

      1. sub-neg N/A

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

      2. associate-+l+ N/A

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

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

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

      4. neg-mul-1 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-out N/A

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

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

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

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

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

      9. exp-lowering-exp.f64 76.8

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

   4. Applied egg-rr 76.8%

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

   5. Taylor expanded in z around 0

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

   7. Simplified 95.4%

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

Alternative 6: 82.5% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-7}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z \cdot \mathsf{fma}\left(z, -0.5, -1\right)}{t}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.1e-7) x (fma y (/ (* z (fma z -0.5 -1.0)) t) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.1e-7) {
		tmp = x;
	} else {
		tmp = fma(y, ((z * fma(z, -0.5, -1.0)) / t), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.1e-7)
		tmp = x;
	else
		tmp = fma(y, Float64(Float64(z * fma(z, -0.5, -1.0)) / t), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.1e-7], x, N[(y * N[(N[(z * N[(z * -0.5 + -1.0), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1e-7

   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 x around inf

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

   5. Simplified 64.1%

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

   if -2.1e-7 < z

   1. Initial program 53.5%

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

   3. Taylor expanded in y around 0

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

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

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

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

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

      3. accelerator-lowering-expm1.f64 86.0

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

   5. Simplified 86.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      8. accelerator-lowering-expm1.f64 88.1

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

   7. Applied egg-rr 88.1%

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

   8. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. div-sub N/A

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

      3. associate-/l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 88.6

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

   10. Simplified 88.6%

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

Alternative 7: 82.4% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.2e-8) x (- x (* y (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.2e-8) {
		tmp = x;
	} else {
		tmp = x - (y * (z / t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-2.2d-8)) then
        tmp = x
    else
        tmp = x - (y * (z / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.2e-8) {
		tmp = x;
	} else {
		tmp = x - (y * (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -2.2e-8:
		tmp = x
	else:
		tmp = x - (y * (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.2e-8)
		tmp = 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 <= -2.2e-8)
		tmp = x;
	else
		tmp = x - (y * (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.2e-8], x, N[(x - N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1999999999999998e-8

   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 x around inf

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

   5. Simplified 64.1%

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

   if -2.1999999999999998e-8 < z

   1. Initial program 53.5%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

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

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

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

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

      9. *-lowering-*.f64 86.5

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

   5. Simplified 86.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 88.6

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

   7. Applied egg-rr 88.6%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.6% accurate, 226.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.7%

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

3. Taylor expanded in x around inf

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

5. Simplified 71.4%

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

Developer Target 1: 74.7% 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 2024195 
(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)))