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

Percentage Accurate: 60.8% → 93.2%

Time: 20.3s

Alternatives: 16

Speedup: 11.3×

• could not determine a ground truth

  1. x = 1.59515588423571e+295
  2. y = 1.2609810703524207e-83
  3. z = 8.260745630433978e+195
  4. t = -1.0404787828068488e+95

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 60.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{t}, \mathsf{log1p}\left(\mathsf{fma}\left(e^{z}, y, -y\right)\right), x\right)\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{-59}:\\ \;\;\;\;x - \frac{y}{\frac{\mathsf{fma}\left(t \cdot z, -0.5, t\right)}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(z, y, 1\right)\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -4.8e-10)
   (fma (/ -1.0 t) (log1p (fma (exp z) y (- y))) x)
   (if (<= z 7.4e-59)
     (- x (/ y (/ (fma (* t z) -0.5 t) z)))
     (- x (/ (log (fma z y 1.0)) t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4.8e-10) {
		tmp = fma((-1.0 / t), log1p(fma(exp(z), y, -y)), x);
	} else if (z <= 7.4e-59) {
		tmp = x - (y / (fma((t * z), -0.5, t) / z));
	} else {
		tmp = x - (log(fma(z, y, 1.0)) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -4.8e-10)
		tmp = fma(Float64(-1.0 / t), log1p(fma(exp(z), y, Float64(-y))), x);
	elseif (z <= 7.4e-59)
		tmp = Float64(x - Float64(y / Float64(fma(Float64(t * z), -0.5, t) / z)));
	else
		tmp = Float64(x - Float64(log(fma(z, y, 1.0)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -4.8e-10], N[(N[(-1.0 / t), $MachinePrecision] * N[Log[1 + N[(N[Exp[z], $MachinePrecision] * y + (-y)), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 7.4e-59], N[(x - N[(y / N[(N[(N[(t * z), $MachinePrecision] * -0.5 + t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(z * y + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.8e-10

   1. Initial program 78.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.4%

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

   if -4.8e-10 < z < 7.3999999999999998e-59

   1. Initial program 57.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 93.1

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

   5. Applied rewrites 93.1%

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

   7. Applied rewrites 93.1%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 93.1%

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

   if 7.3999999999999998e-59 < z

   1. Initial program 62.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 98.1

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

   5. Applied rewrites 98.1%

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

Alternative 2: 93.3% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ (* y (exp z)) (- 1.0 y)) 2.0)
   (- x (/ y (/ t (expm1 z))))
   (- x (/ (log (* (expm1 z) y)) t))))

C

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

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((y * Math.exp(z)) + (1.0 - y)) <= 2.0) {
		tmp = x - (y / (t / Math.expm1(z)));
	} else {
		tmp = x - (Math.log((Math.expm1(z) * y)) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((y * math.exp(z)) + (1.0 - y)) <= 2.0:
		tmp = x - (y / (t / math.expm1(z)))
	else:
		tmp = x - (math.log((math.expm1(z) * y)) / t)
	return tmp

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 60.8%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 92.9

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

   5. Applied rewrites 92.9%

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

   7. Applied rewrites 93.0%

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

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

   1. Initial program 95.1%

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

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 96.3

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

   5. Applied rewrites 96.3%

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

4. Final simplification 93.3%

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

Alternative 3: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+177}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(z\right) \cdot y\right)}{-t}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+180}:\\ \;\;\;\;x - \frac{1}{\frac{\mathsf{fma}\left(y \cdot t, 0.5, \frac{t}{\mathsf{expm1}\left(z\right)}\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(z, y, 1\right)\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -5.2e+177)
   (/ (log1p (* (expm1 z) y)) (- t))
   (if (<= y 4.6e+180)
     (- x (/ 1.0 (/ (fma (* y t) 0.5 (/ t (expm1 z))) y)))
     (- x (/ (log (fma z y 1.0)) t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5.2e+177) {
		tmp = log1p((expm1(z) * y)) / -t;
	} else if (y <= 4.6e+180) {
		tmp = x - (1.0 / (fma((y * t), 0.5, (t / expm1(z))) / y));
	} else {
		tmp = x - (log(fma(z, y, 1.0)) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -5.2e+177)
		tmp = Float64(log1p(Float64(expm1(z) * y)) / Float64(-t));
	elseif (y <= 4.6e+180)
		tmp = Float64(x - Float64(1.0 / Float64(fma(Float64(y * t), 0.5, Float64(t / expm1(z))) / y)));
	else
		tmp = Float64(x - Float64(log(fma(z, y, 1.0)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -5.2e+177], N[(N[Log[1 + N[(N[(Exp[z] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] / (-t)), $MachinePrecision], If[LessEqual[y, 4.6e+180], N[(x - N[(1.0 / N[(N[(N[(y * t), $MachinePrecision] * 0.5 + N[(t / N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(z * y + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.19999999999999959e177

   1. Initial program 63.9%

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

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

      11. lower-*.f64 N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-neg.f64 73.0

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

   5. Applied rewrites 73.0%

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

   if -5.19999999999999959e177 < y < 4.5999999999999998e180

   1. Initial program 68.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 93.4

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

   5. Applied rewrites 93.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 93.3

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

   7. Applied rewrites 93.3%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 95.0

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

   10. Applied rewrites 95.0%

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

   if 4.5999999999999998e180 < y

   1. Initial program 7.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+177}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(z\right) \cdot y\right)}{-t}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+180}:\\ \;\;\;\;x - \frac{1}{\frac{\mathsf{fma}\left(y \cdot t, 0.5, \frac{t}{\mathsf{expm1}\left(z\right)}\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(z, y, 1\right)\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{+180}:\\ \;\;\;\;x - \frac{1}{\frac{\mathsf{fma}\left(y \cdot t, 0.5, \frac{t}{\mathsf{expm1}\left(z\right)}\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(z, y, 1\right)\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 4.6e+180)
   (- x (/ 1.0 (/ (fma (* y t) 0.5 (/ t (expm1 z))) y)))
   (- x (/ (log (fma z y 1.0)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.6e+180) {
		tmp = x - (1.0 / (fma((y * t), 0.5, (t / expm1(z))) / y));
	} else {
		tmp = x - (log(fma(z, y, 1.0)) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 4.6e+180)
		tmp = Float64(x - Float64(1.0 / Float64(fma(Float64(y * t), 0.5, Float64(t / expm1(z))) / y)));
	else
		tmp = Float64(x - Float64(log(fma(z, y, 1.0)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 4.6e+180], N[(x - N[(1.0 / N[(N[(N[(y * t), $MachinePrecision] * 0.5 + N[(t / N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(z * y + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.5999999999999998e180

   1. Initial program 68.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 88.0

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

   5. Applied rewrites 88.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 88.0

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

   7. Applied rewrites 88.0%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 89.0

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

   10. Applied rewrites 89.0%

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

   if 4.5999999999999998e180 < y

   1. Initial program 7.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{+180}:\\ \;\;\;\;x - \frac{1}{\frac{\mathsf{fma}\left(y \cdot t, 0.5, \frac{t}{\mathsf{expm1}\left(z\right)}\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(z, y, 1\right)\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((y * exp(z)) + (1.0 - y)) <= 0.0) {
		tmp = x - ((1.0 / t) * (y * z));
	} else {
		tmp = x - (z / (t / y));
	}
	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 (((y * exp(z)) + (1.0d0 - y)) <= 0.0d0) then
        tmp = x - ((1.0d0 / t) * (y * z))
    else
        tmp = x - (z / (t / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((y * Math.exp(z)) + (1.0 - y)) <= 0.0) {
		tmp = x - ((1.0 / t) * (y * z));
	} else {
		tmp = x - (z / (t / y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((y * math.exp(z)) + (1.0 - y)) <= 0.0:
		tmp = x - ((1.0 / t) * (y * z))
	else:
		tmp = x - (z / (t / y))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision] + N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.0], N[(x - N[(N[(1.0 / t), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(z / N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 2.6%

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

   3. Taylor expanded in z around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 69.6

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

   5. Applied rewrites 69.6%

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

   7. Applied rewrites 79.7%

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

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

   1. Initial program 84.9%

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

   3. Taylor expanded in z around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 76.1

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

   5. Applied rewrites 76.1%

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

   7. Applied rewrites 76.6%

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

4. Final simplification 77.4%

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

Alternative 6: 88.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{\log \left(\mathsf{fma}\left(z, y, 1\right)\right)}{t}\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{+85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+180}:\\ \;\;\;\;x - \frac{y}{\frac{t}{\mathsf{expm1}\left(z\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ (log (fma z y 1.0)) t))))
   (if (<= y -8.5e+85)
     t_1
     (if (<= y 3.3e+180) (- x (/ y (/ t (expm1 z)))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (log(fma(z, y, 1.0)) / t);
	double tmp;
	if (y <= -8.5e+85) {
		tmp = t_1;
	} else if (y <= 3.3e+180) {
		tmp = x - (y / (t / expm1(z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(log(fma(z, y, 1.0)) / t))
	tmp = 0.0
	if (y <= -8.5e+85)
		tmp = t_1;
	elseif (y <= 3.3e+180)
		tmp = Float64(x - Float64(y / Float64(t / expm1(z))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(N[Log[N[(z * y + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8.5e+85], t$95$1, If[LessEqual[y, 3.3e+180], N[(x - N[(y / N[(t / N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.4999999999999994e85 or 3.29999999999999989e180 < y

   1. Initial program 35.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 65.9

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

   5. Applied rewrites 65.9%

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

   if -8.4999999999999994e85 < y < 3.29999999999999989e180

   1. Initial program 72.2%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 97.4

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

   5. Applied rewrites 97.4%

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

   7. Applied rewrites 97.5%

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

Alternative 7: 88.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{\log \left(\mathsf{fma}\left(z, y, 1\right)\right)}{t}\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{+85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+180}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ (log (fma z y 1.0)) t))))
   (if (<= y -8.5e+85)
     t_1
     (if (<= y 3.3e+180) (- x (* (/ (expm1 z) t) y)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (log(fma(z, y, 1.0)) / t);
	double tmp;
	if (y <= -8.5e+85) {
		tmp = t_1;
	} else if (y <= 3.3e+180) {
		tmp = x - ((expm1(z) / t) * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(log(fma(z, y, 1.0)) / t))
	tmp = 0.0
	if (y <= -8.5e+85)
		tmp = t_1;
	elseif (y <= 3.3e+180)
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(N[Log[N[(z * y + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8.5e+85], t$95$1, If[LessEqual[y, 3.3e+180], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.4999999999999994e85 or 3.29999999999999989e180 < y

   1. Initial program 35.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 65.9

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

   5. Applied rewrites 65.9%

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

   if -8.4999999999999994e85 < y < 3.29999999999999989e180

   1. Initial program 72.2%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 97.4

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

   5. Applied rewrites 97.4%

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

Alternative 8: 86.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.25 \cdot 10^{+29}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log 1}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 1.25e+29) (- x (* (/ (expm1 z) t) y)) (- x (/ (log 1.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.25e+29) {
		tmp = x - ((expm1(z) / t) * y);
	} else {
		tmp = x - (log(1.0) / t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.25e+29) {
		tmp = x - ((Math.expm1(z) / t) * y);
	} else {
		tmp = x - (Math.log(1.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 1.25e+29:
		tmp = x - ((math.expm1(z) / t) * y)
	else:
		tmp = x - (math.log(1.0) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 1.25e+29)
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	else
		tmp = Float64(x - Float64(log(1.0) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 1.25e+29], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[1.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.25e29

   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 y around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 85.4

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

   5. Applied rewrites 85.4%

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

   if 1.25e29 < t

   1. Initial program 72.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 97.1%

      \[\leadsto x - \frac{\log \color{blue}{1}}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 82.3% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ x - \frac{y}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, t, 0.08333333333333333 \cdot \left(t \cdot z\right)\right), z, t\right)}{z}} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- x (/ y (/ (fma (fma -0.5 t (* 0.08333333333333333 (* t z))) z t) z))))

C

double code(double x, double y, double z, double t) {
	return x - (y / (fma(fma(-0.5, t, (0.08333333333333333 * (t * z))), z, t) / z));
}

Julia

function code(x, y, z, t)
	return Float64(x - Float64(y / Float64(fma(fma(-0.5, t, Float64(0.08333333333333333 * Float64(t * z))), z, t) / z)))
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(y / N[(N[(N[(-0.5 * t + N[(0.08333333333333333 * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z + t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.3%

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

3. Taylor expanded in y around 0

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

   1. associate-/l* N/A

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

   2. div-sub N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. div-sub N/A

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

   6. lower-/.f64 N/A

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

   7. lower-expm1.f64 85.9

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

5. Applied rewrites 85.9%

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

7. Applied rewrites 86.0%

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

8. Taylor expanded in z around 0

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

10. Applied rewrites 81.8%

    \[\leadsto x - \frac{y}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, t, 0.08333333333333333 \cdot \left(t \cdot z\right)\right), z, t\right)}{\color{blue}{z}}} \]
11. Add Preprocessing

Alternative 10: 82.2% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ x - \frac{y}{\frac{\mathsf{fma}\left(t \cdot z, -0.5, t\right)}{z}} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- x (/ y (/ (fma (* t z) -0.5 t) z))))

C

double code(double x, double y, double z, double t) {
	return x - (y / (fma((t * z), -0.5, t) / z));
}

Julia

function code(x, y, z, t)
	return Float64(x - Float64(y / Float64(fma(Float64(t * z), -0.5, t) / z)))
end

Wolfram

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

Derivation

1. Initial program 64.3%

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

3. Taylor expanded in y around 0

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

   1. associate-/l* N/A

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

   2. div-sub N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. div-sub N/A

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

   6. lower-/.f64 N/A

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

   7. lower-expm1.f64 85.9

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

5. Applied rewrites 85.9%

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

7. Applied rewrites 86.0%

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

8. Taylor expanded in z around 0

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

10. Applied rewrites 81.6%

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

Alternative 11: 73.9% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

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 - ((1.0d0 / t) * (y * z))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - ((1.0 / t) * (y * z));
}

Python

def code(x, y, z, t):
	return x - ((1.0 / t) * (y * z))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

3. Taylor expanded in z around 0

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 74.5

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

5. Applied rewrites 74.5%

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

7. Applied rewrites 75.8%

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

8. Final simplification 75.8%

   \[\leadsto x - \frac{1}{t} \cdot \left(y \cdot z\right) \]
9. Add Preprocessing

Alternative 12: 74.0% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

3. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 75.8

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

5. Applied rewrites 75.8%

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

6. Final simplification 75.8%

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

Alternative 13: 74.4% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

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 - ((z / t) * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

3. Taylor expanded in y around 0

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

   1. associate-/l* N/A

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

   2. div-sub N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. div-sub N/A

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

   6. lower-/.f64 N/A

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

   7. lower-expm1.f64 85.9

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

5. Applied rewrites 85.9%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 75.2%

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

Alternative 14: 14.6% accurate, 11.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

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

   11. lower-*.f64 N/A

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

   12. lower-expm1.f64 N/A

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

   13. lower-neg.f64 27.4

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

5. Applied rewrites 27.4%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 12.8%

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

9. Final simplification 12.8%

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

Alternative 15: 15.2% accurate, 11.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

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

   11. lower-*.f64 N/A

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

   12. lower-expm1.f64 N/A

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

   13. lower-neg.f64 27.4

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

5. Applied rewrites 27.4%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 12.8%

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

10. Applied rewrites 7.8%

    \[\leadsto \frac{z \cdot y}{\frac{0 - t \cdot t}{\color{blue}{0 + t}}} \]

11. Taylor expanded in z around 0

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

13. Applied rewrites 12.5%

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

14. Final simplification 12.5%

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

Alternative 16: 13.5% accurate, 11.9× speedup

Math

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

FPCore

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

C

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

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 = (-y / t) * z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

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

   11. lower-*.f64 N/A

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

   12. lower-expm1.f64 N/A

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

   13. lower-neg.f64 27.4

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

5. Applied rewrites 27.4%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 12.8%

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

9. Taylor expanded in z around 0

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

11. Applied rewrites 11.1%

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

12. Final simplification 11.1%

    \[\leadsto \frac{-y}{t} \cdot z \]
13. 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 2024235 
(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)))