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

Percentage Accurate: 62.0% → 98.4%

Time: 21.3s

Alternatives: 9

Speedup: 226.0×

• could not determine a ground truth

  1. x = 2.7734940734382014e-185
  2. y = 4.481771133453665e+76
  3. z = 1.068316529789612e+100
  4. t = -5.654817660597495e-296

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: 62.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% 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 59.2%

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

3. Taylor expanded in z around inf

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

   1. sub-neg N/A

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

   2. associate-+l+ N/A

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

   3. *-rgt-identity N/A

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

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

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

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

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

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

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

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

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

   8. accelerator-lowering-expm1.f64 98.4

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

5. Simplified 98.4%

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

Alternative 2: 94.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1 - y\right) + y \cdot e^{z}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+34}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+182}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{-t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{\frac{\mathsf{fma}\left(0.5, y \cdot t, \frac{t}{z}\right)}{y}}\\ \end{array} \end{array} \]

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 2e+34)
       (- x (* y (/ (expm1 z) t)))
       (if (<= t_1 4e+182)
         (/ (log1p (* y (expm1 z))) (- t))
         (+ x (/ -1.0 (/ (fma 0.5 (* y t) (/ t z)) y))))))))

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 <= 2e+34) {
		tmp = x - (y * (expm1(z) / t));
	} else if (t_1 <= 4e+182) {
		tmp = log1p((y * expm1(z))) / -t;
	} else {
		tmp = x + (-1.0 / (fma(0.5, (y * t), (t / z)) / y));
	}
	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 <= 2e+34)
		tmp = Float64(x - Float64(y * Float64(expm1(z) / t)));
	elseif (t_1 <= 4e+182)
		tmp = Float64(log1p(Float64(y * expm1(z))) / Float64(-t));
	else
		tmp = Float64(x + Float64(-1.0 / Float64(fma(0.5, Float64(y * t), Float64(t / z)) / y)));
	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, 2e+34], N[(x - N[(y * N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+182], N[(N[Log[1 + N[(y * N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t)), $MachinePrecision], N[(x + N[(-1.0 / N[(N[(0.5 * N[(y * t), $MachinePrecision] + N[(t / z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 2.1%

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

   3. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. associate-+l+ N/A

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

      3. *-rgt-identity N/A

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

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

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

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

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

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

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

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

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

      8. accelerator-lowering-expm1.f64 99.9

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

   5. Simplified 99.9%

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

   6. Taylor expanded in z around 0

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

      1. *-lowering-*.f64 99.9

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

   8. Simplified 99.9%

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

   if 0.0 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))) < 1.99999999999999989e34

   1. Initial program 80.7%

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

   3. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. associate-+l+ N/A

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

      3. *-rgt-identity N/A

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

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

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

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

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

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

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

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

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

      8. accelerator-lowering-expm1.f64 98.1

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

   5. Simplified 98.1%

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

   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. *-lowering-*.f64 N/A

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

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

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

      4. accelerator-lowering-expm1.f64 98.3

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

   8. Simplified 98.3%

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

   if 1.99999999999999989e34 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))) < 4.0000000000000003e182

   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 x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

      4. sub-neg N/A

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

      5. associate-+l+ N/A

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

      6. *-rgt-identity N/A

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

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

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

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

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

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

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

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

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

      11. accelerator-lowering-expm1.f64 N/A

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

      12. neg-lowering-neg.f64 78.6

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

   5. Simplified 78.6%

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

   if 4.0000000000000003e182 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))

   1. Initial program 86.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 18.0

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

   5. Simplified 18.0%

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

      1. clear-num N/A

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

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

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

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

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

      4. associate-+l- N/A

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

      5. sub-neg N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. accelerator-lowering-fma.f64 18.0

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

   7. Applied egg-rr 18.0%

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

   8. Taylor expanded in y around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. /-lowering-/.f64 75.3

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

   10. Simplified 75.3%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - y\right) + y \cdot e^{z} \leq 0:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \mathbf{elif}\;\left(1 - y\right) + y \cdot e^{z} \leq 2 \cdot 10^{+34}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \mathbf{elif}\;\left(1 - y\right) + y \cdot e^{z} \leq 4 \cdot 10^{+182}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{-t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{\frac{\mathsf{fma}\left(0.5, y \cdot t, \frac{t}{z}\right)}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.7% accurate, 0.6× speedup

Math

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

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 10.0)
       (- x (* y (/ (expm1 z) t)))
       (+ x (/ -1.0 (/ (fma 0.5 (* y t) (/ t z)) y)))))))

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 <= 10.0) {
		tmp = x - (y * (expm1(z) / t));
	} else {
		tmp = x + (-1.0 / (fma(0.5, (y * t), (t / z)) / y));
	}
	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 <= 10.0)
		tmp = Float64(x - Float64(y * Float64(expm1(z) / t)));
	else
		tmp = Float64(x + Float64(-1.0 / Float64(fma(0.5, Float64(y * t), Float64(t / z)) / y)));
	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, 10.0], N[(x - N[(y * N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(-1.0 / N[(N[(0.5 * N[(y * t), $MachinePrecision] + N[(t / z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 2.1%

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

   3. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. associate-+l+ N/A

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

      3. *-rgt-identity N/A

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

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

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

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

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

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

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

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

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

      8. accelerator-lowering-expm1.f64 99.9

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

   5. Simplified 99.9%

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

   6. Taylor expanded in z around 0

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

      1. *-lowering-*.f64 99.9

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

   8. Simplified 99.9%

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

   if 0.0 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))) < 10

   1. Initial program 80.3%

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

   3. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. associate-+l+ N/A

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

      3. *-rgt-identity N/A

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

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

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

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

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

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

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

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

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

      8. accelerator-lowering-expm1.f64 98.1

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

   5. Simplified 98.1%

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

   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. *-lowering-*.f64 N/A

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

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

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

      4. accelerator-lowering-expm1.f64 99.1

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

   8. Simplified 99.1%

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

   if 10 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))

   1. Initial program 92.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 34.2

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

   5. Simplified 34.2%

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

      1. clear-num N/A

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

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

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

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

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

      4. associate-+l- N/A

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

      5. sub-neg N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. accelerator-lowering-fma.f64 34.2

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

   7. Applied egg-rr 34.2%

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

   8. Taylor expanded in y around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. /-lowering-/.f64 51.6

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

   10. Simplified 51.6%

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

4. Final simplification 94.7%

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

Alternative 4: 87.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-64}:\\ \;\;\;\;x + \frac{-1}{\frac{\mathsf{fma}\left(0.5, y \cdot t, \frac{t}{z}\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2e-64)
   (+ x (/ -1.0 (/ (fma 0.5 (* y t) (/ t z)) y)))
   (- x (* y (/ (expm1 z) t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2e-64) {
		tmp = x + (-1.0 / (fma(0.5, (y * t), (t / z)) / y));
	} else {
		tmp = x - (y * (expm1(z) / t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2e-64)
		tmp = Float64(x + Float64(-1.0 / Float64(fma(0.5, Float64(y * t), Float64(t / z)) / y)));
	else
		tmp = Float64(x - Float64(y * Float64(expm1(z) / t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -2e-64], N[(x + N[(-1.0 / N[(N[(0.5 * N[(y * t), $MachinePrecision] + N[(t / z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $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 < -1.99999999999999993e-64

   1. Initial program 47.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 30.7

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

   5. Simplified 30.7%

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

      1. clear-num N/A

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

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

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

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

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

      4. associate-+l- N/A

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

      5. sub-neg N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. accelerator-lowering-fma.f64 60.1

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

   7. Applied egg-rr 60.1%

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

   8. Taylor expanded in y around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. /-lowering-/.f64 79.7

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

   10. Simplified 79.7%

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

   if -1.99999999999999993e-64 < y

   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 z around inf

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

      1. sub-neg N/A

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

      2. associate-+l+ N/A

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

      3. *-rgt-identity N/A

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

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

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

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

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

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

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

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

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

      8. accelerator-lowering-expm1.f64 98.2

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

   5. Simplified 98.2%

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

   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. *-lowering-*.f64 N/A

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

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

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

      4. accelerator-lowering-expm1.f64 92.7

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

   8. Simplified 92.7%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-64}:\\ \;\;\;\;x + \frac{-1}{\frac{\mathsf{fma}\left(0.5, y \cdot t, \frac{t}{z}\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 4.4 \cdot 10^{-116}:\\ \;\;\;\;x + \frac{-1}{\frac{\mathsf{fma}\left(0.5, y \cdot t, \frac{t}{z}\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{\frac{\mathsf{fma}\left(0.5, z \cdot t, \frac{t}{y}\right)}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 4.4e-116)
   (+ x (/ -1.0 (/ (fma 0.5 (* y t) (/ t z)) y)))
   (+ x (/ -1.0 (/ (fma 0.5 (* z t) (/ t y)) z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 4.4e-116) {
		tmp = x + (-1.0 / (fma(0.5, (y * t), (t / z)) / y));
	} else {
		tmp = x + (-1.0 / (fma(0.5, (z * t), (t / y)) / z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 4.4e-116)
		tmp = Float64(x + Float64(-1.0 / Float64(fma(0.5, Float64(y * t), Float64(t / z)) / y)));
	else
		tmp = Float64(x + Float64(-1.0 / Float64(fma(0.5, Float64(z * t), Float64(t / y)) / z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 4.4e-116], N[(x + N[(-1.0 / N[(N[(0.5 * N[(y * t), $MachinePrecision] + N[(t / z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(-1.0 / N[(N[(0.5 * N[(z * t), $MachinePrecision] + N[(t / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.4000000000000002e-116

   1. Initial program 59.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 51.0

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

   5. Simplified 51.0%

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

      1. clear-num N/A

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

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

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

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

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

      4. associate-+l- N/A

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

      5. sub-neg N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. accelerator-lowering-fma.f64 64.0

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

   7. Applied egg-rr 64.0%

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

   8. Taylor expanded in y around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. /-lowering-/.f64 82.1

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

   10. Simplified 82.1%

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

   if 4.4000000000000002e-116 < t

   1. Initial program 58.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 46.9

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

   5. Simplified 46.9%

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

      1. clear-num N/A

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

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

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

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

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

      4. associate-+l- N/A

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

      5. sub-neg N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. accelerator-lowering-fma.f64 68.3

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

   7. Applied egg-rr 68.3%

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

   8. Taylor expanded in z around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. /-lowering-/.f64 85.8

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

   10. Simplified 85.8%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.4 \cdot 10^{-116}:\\ \;\;\;\;x + \frac{-1}{\frac{\mathsf{fma}\left(0.5, y \cdot t, \frac{t}{z}\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{\frac{\mathsf{fma}\left(0.5, z \cdot t, \frac{t}{y}\right)}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-299}:\\ \;\;\;\;x + \frac{-1}{\frac{\mathsf{fma}\left(0.5, y \cdot t, \frac{t}{z}\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.5e-299)
   (+ x (/ -1.0 (/ (fma 0.5 (* y t) (/ t z)) y)))
   (- x (/ (* y z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.5e-299) {
		tmp = x + (-1.0 / (fma(0.5, (y * t), (t / z)) / y));
	} else {
		tmp = x - ((y * z) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.5e-299)
		tmp = Float64(x + Float64(-1.0 / Float64(fma(0.5, Float64(y * t), Float64(t / z)) / y)));
	else
		tmp = Float64(x - Float64(Float64(y * z) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.5e-299], N[(x + N[(-1.0 / N[(N[(0.5 * N[(y * t), $MachinePrecision] + N[(t / z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.49999999999999992e-299

   1. Initial program 63.1%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 52.8

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

   5. Simplified 52.8%

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

      1. clear-num N/A

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

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

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

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

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

      4. associate-+l- N/A

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

      5. sub-neg N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. accelerator-lowering-fma.f64 68.7

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

   7. Applied egg-rr 68.7%

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

   8. Taylor expanded in y around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. /-lowering-/.f64 82.9

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

   10. Simplified 82.9%

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

   if 1.49999999999999992e-299 < y

   1. Initial program 53.8%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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. *-lowering-*.f64 82.8

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

   5. Simplified 82.8%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-299}:\\ \;\;\;\;x + \frac{-1}{\frac{\mathsf{fma}\left(0.5, y \cdot t, \frac{t}{z}\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.3% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{\mathsf{fma}\left(z, z \cdot 0.5, z\right)}{t}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -8.8e-8) x (fma (- y) (/ (fma z (* z 0.5) z) t) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -8.8e-8) {
		tmp = x;
	} else {
		tmp = fma(-y, (fma(z, (z * 0.5), z) / t), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -8.8e-8)
		tmp = x;
	else
		tmp = fma(Float64(-y), Float64(fma(z, Float64(z * 0.5), z) / t), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -8.8e-8], x, N[((-y) * N[(N[(z * N[(z * 0.5), $MachinePrecision] + z), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.7999999999999994e-8

   1. Initial program 82.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 67.5%

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

   if -8.7999999999999994e-8 < z

   1. Initial program 50.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 79.9

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

   5. Simplified 79.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

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

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

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

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

      10. +-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. *-rgt-identity N/A

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 88.4

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

   8. Simplified 88.4%

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

Alternative 8: 82.3% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -8.8e-8) x (- x (* y (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -8.8e-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 <= (-8.8d-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 <= -8.8e-8) {
		tmp = x;
	} else {
		tmp = x - (y * (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -8.8e-8:
		tmp = x
	else:
		tmp = x - (y * (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -8.8e-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 <= -8.8e-8)
		tmp = x;
	else
		tmp = x - (y * (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -8.8e-8], x, N[(x - N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.7999999999999994e-8

   1. Initial program 82.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 67.5%

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

   if -8.7999999999999994e-8 < z

   1. Initial program 50.7%

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

   3. Taylor expanded in z around 0

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

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

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

      2. +-commutative N/A

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

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

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

   5. Simplified 72.1%

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

   6. Taylor expanded in z around 0

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

   8. Simplified 86.9%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 88.2

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

   10. Applied egg-rr 88.2%

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

Alternative 9: 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 59.2%

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

3. Taylor expanded in x around inf

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

5. Simplified 70.0%

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

Developer Target 1: 75.0% 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 2024204 
(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)))