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

Percentage Accurate: 61.4% → 98.6%

Time: 21.8s

Alternatives: 15

Speedup: 226.0×

• could not determine a ground truth

  1. x = 1.1017352908958928e-276
  2. y = 2.319995456742995e+238
  3. z = 8.124616209561728e+151
  4. t = 1.4525796371024298e-233

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.4% 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: 98.6% 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 60.9%

   \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
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 99.6

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

5. Simplified 99.6%

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

Alternative 2: 94.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1 - y\right) + y \cdot e^{z}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \mathbf{elif}\;t\_1 \leq 50000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{expm1}\left(z\right)}{t}, -y, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+117}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{-t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(t, \frac{x}{\mathsf{expm1}\left(z\right)}, 0.5 \cdot \left(y \cdot \left(x \cdot t\right)\right)\right)}{y}}, -x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (- 1.0 y) (* y (exp z)))))
   (if (<= t_1 0.0)
     (- x (/ (log1p (* y z)) t))
     (if (<= t_1 50000000.0)
       (fma (/ (expm1 z) t) (- y) x)
       (if (<= t_1 1e+117)
         (/ (log1p (* y (expm1 z))) (- t))
         (fma
          (/ 1.0 (/ (fma t (/ x (expm1 z)) (* 0.5 (* y (* x t)))) y))
          (- x)
          x))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (1.0 - y) + (y * exp(z));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = x - (log1p((y * z)) / t);
	} else if (t_1 <= 50000000.0) {
		tmp = fma((expm1(z) / t), -y, x);
	} else if (t_1 <= 1e+117) {
		tmp = log1p((y * expm1(z))) / -t;
	} else {
		tmp = fma((1.0 / (fma(t, (x / expm1(z)), (0.5 * (y * (x * t)))) / y)), -x, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(1.0 - y) + Float64(y * exp(z)))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(x - Float64(log1p(Float64(y * z)) / t));
	elseif (t_1 <= 50000000.0)
		tmp = fma(Float64(expm1(z) / t), Float64(-y), x);
	elseif (t_1 <= 1e+117)
		tmp = Float64(log1p(Float64(y * expm1(z))) / Float64(-t));
	else
		tmp = fma(Float64(1.0 / Float64(fma(t, Float64(x / expm1(z)), Float64(0.5 * Float64(y * Float64(x * t)))) / y)), Float64(-x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(1.0 - y), $MachinePrecision] + N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(x - N[(N[Log[1 + N[(y * z), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 50000000.0], N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * (-y) + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+117], N[(N[Log[1 + N[(y * N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t)), $MachinePrecision], N[(N[(1.0 / N[(N[(t * N[(x / N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(y * N[(x * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * (-x) + x), $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 2.3%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 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))) < 5e7

   1. Initial program 78.1%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 99.3

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

   5. Simplified 99.3%

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

   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. div-sub N/A

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

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

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

      4. div-sub N/A

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

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

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

      6. accelerator-lowering-expm1.f64 99.4

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

   8. Simplified 99.4%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

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

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

      8. neg-lowering-neg.f64 99.4

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

   10. Applied egg-rr 99.4%

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

   if 5e7 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))) < 1.00000000000000005e117

   1. Initial program 97.7%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 76.4

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

   5. Simplified 76.4%

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

   if 1.00000000000000005e117 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

   5. Simplified 92.4%

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

      1. associate-/r* N/A

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

      2. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

      8. accelerator-lowering-expm1.f64 92.4

         \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{x}{\frac{\mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(z\right)}\right)}{t}}}, -x, x\right) \]

   7. Applied egg-rr 92.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{x}{\frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}}}}, -x, x\right) \]

   8. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

         \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(t, \frac{x}{\color{blue}{\mathsf{expm1}\left(z\right)}}, \frac{1}{2} \cdot \left(t \cdot \left(x \cdot y\right)\right)\right)}{y}}, \mathsf{neg}\left(x\right), x\right) \]

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

         \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(t, \frac{x}{\mathsf{expm1}\left(z\right)}, \color{blue}{\frac{1}{2} \cdot \left(t \cdot \left(x \cdot y\right)\right)}\right)}{y}}, \mathsf{neg}\left(x\right), x\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(t, \frac{x}{\mathsf{expm1}\left(z\right)}, \frac{1}{2} \cdot \color{blue}{\left(\left(x \cdot y\right) \cdot t\right)}\right)}{y}}, \mathsf{neg}\left(x\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(t, \frac{x}{\mathsf{expm1}\left(z\right)}, \frac{1}{2} \cdot \left(\color{blue}{\left(y \cdot x\right)} \cdot t\right)\right)}{y}}, \mathsf{neg}\left(x\right), x\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(t, \frac{x}{\mathsf{expm1}\left(z\right)}, \frac{1}{2} \cdot \color{blue}{\left(y \cdot \left(x \cdot t\right)\right)}\right)}{y}}, \mathsf{neg}\left(x\right), x\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(t, \frac{x}{\mathsf{expm1}\left(z\right)}, \frac{1}{2} \cdot \left(y \cdot \color{blue}{\left(t \cdot x\right)}\right)\right)}{y}}, \mathsf{neg}\left(x\right), x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(t, \frac{x}{\mathsf{expm1}\left(z\right)}, \frac{1}{2} \cdot \color{blue}{\left(y \cdot \left(t \cdot x\right)\right)}\right)}{y}}, \mathsf{neg}\left(x\right), x\right) \]

      13. *-lowering-*.f64 68.5

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

   10. Simplified 68.5%

       \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{\frac{\mathsf{fma}\left(t, \frac{x}{\mathsf{expm1}\left(z\right)}, 0.5 \cdot \left(y \cdot \left(t \cdot x\right)\right)\right)}{y}}}, -x, x\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - y\right) + y \cdot e^{z} \leq 0:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \mathbf{elif}\;\left(1 - y\right) + y \cdot e^{z} \leq 50000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{expm1}\left(z\right)}{t}, -y, x\right)\\ \mathbf{elif}\;\left(1 - y\right) + y \cdot e^{z} \leq 10^{+117}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{-t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(t, \frac{x}{\mathsf{expm1}\left(z\right)}, 0.5 \cdot \left(y \cdot \left(x \cdot t\right)\right)\right)}{y}}, -x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(1.0 - y), $MachinePrecision] + N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(x - N[(N[Log[1 + N[(y * z), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * (-y) + x), $MachinePrecision], N[(N[(1.0 / N[(N[(t * N[(x / N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(y * N[(x * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * (-x) + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 2.3%

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

   1. Initial program 77.7%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 99.3

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

   5. Simplified 99.3%

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

   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. div-sub N/A

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

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

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

      4. div-sub N/A

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

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

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

      6. accelerator-lowering-expm1.f64 99.9

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

   8. Simplified 99.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

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

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

      8. neg-lowering-neg.f64 99.9

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

   10. Applied egg-rr 99.9%

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

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

   1. Initial program 98.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

   5. Simplified 86.9%

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

      1. associate-/r* N/A

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

      2. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

      8. accelerator-lowering-expm1.f64 87.0

         \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{x}{\frac{\mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(z\right)}\right)}{t}}}, -x, x\right) \]

   7. Applied egg-rr 87.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{x}{\frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}}}}, -x, x\right) \]

   8. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

         \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(t, \frac{x}{\color{blue}{\mathsf{expm1}\left(z\right)}}, \frac{1}{2} \cdot \left(t \cdot \left(x \cdot y\right)\right)\right)}{y}}, \mathsf{neg}\left(x\right), x\right) \]

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

         \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(t, \frac{x}{\mathsf{expm1}\left(z\right)}, \color{blue}{\frac{1}{2} \cdot \left(t \cdot \left(x \cdot y\right)\right)}\right)}{y}}, \mathsf{neg}\left(x\right), x\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(t, \frac{x}{\mathsf{expm1}\left(z\right)}, \frac{1}{2} \cdot \color{blue}{\left(\left(x \cdot y\right) \cdot t\right)}\right)}{y}}, \mathsf{neg}\left(x\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(t, \frac{x}{\mathsf{expm1}\left(z\right)}, \frac{1}{2} \cdot \left(\color{blue}{\left(y \cdot x\right)} \cdot t\right)\right)}{y}}, \mathsf{neg}\left(x\right), x\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(t, \frac{x}{\mathsf{expm1}\left(z\right)}, \frac{1}{2} \cdot \color{blue}{\left(y \cdot \left(x \cdot t\right)\right)}\right)}{y}}, \mathsf{neg}\left(x\right), x\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(t, \frac{x}{\mathsf{expm1}\left(z\right)}, \frac{1}{2} \cdot \left(y \cdot \color{blue}{\left(t \cdot x\right)}\right)\right)}{y}}, \mathsf{neg}\left(x\right), x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(t, \frac{x}{\mathsf{expm1}\left(z\right)}, \frac{1}{2} \cdot \color{blue}{\left(y \cdot \left(t \cdot x\right)\right)}\right)}{y}}, \mathsf{neg}\left(x\right), x\right) \]

      13. *-lowering-*.f64 55.2

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

   10. Simplified 55.2%

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

4. Final simplification 92.4%

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

Alternative 4: 94.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1 - y\right) + y \cdot e^{z}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \mathbf{elif}\;t\_1 \leq 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{expm1}\left(z\right)}{t}, -y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (- 1.0 y) (* y (exp z)))))
   (if (<= t_1 0.0)
     (- x (/ (log1p (* y z)) t))
     (if (<= t_1 1e+63) (fma (/ (expm1 z) t) (- y) x) x))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (1.0 - y) + (y * exp(z));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = x - (log1p((y * z)) / t);
	} else if (t_1 <= 1e+63) {
		tmp = fma((expm1(z) / t), -y, x);
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(1.0 - y) + Float64(y * exp(z)))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(x - Float64(log1p(Float64(y * z)) / t));
	elseif (t_1 <= 1e+63)
		tmp = fma(Float64(expm1(z) / t), Float64(-y), x);
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(1.0 - y), $MachinePrecision] + N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(x - N[(N[Log[1 + N[(y * z), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+63], N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * (-y) + x), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 2.3%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 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))) < 1.00000000000000006e63

   1. Initial program 79.3%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 99.4

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

   5. Simplified 99.4%

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

   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. div-sub N/A

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

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

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

      4. div-sub N/A

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

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

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

      6. accelerator-lowering-expm1.f64 95.3

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

   8. Simplified 95.3%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

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

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

      8. neg-lowering-neg.f64 95.3

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

   10. Applied egg-rr 95.3%

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

   if 1.00000000000000006e63 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))

   1. Initial program 98.3%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 56.2%

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

Alternative 5: 94.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1 - y\right) + y \cdot e^{z}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \mathbf{elif}\;t\_1 \leq 10^{+63}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (- 1.0 y) (* y (exp z)))))
   (if (<= t_1 0.0)
     (- x (/ (log1p (* y z)) t))
     (if (<= t_1 1e+63) (- x (* y (/ (expm1 z) t))) x))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (1.0 - y) + (y * exp(z));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = x - (log1p((y * z)) / t);
	} else if (t_1 <= 1e+63) {
		tmp = x - (y * (expm1(z) / t));
	} else {
		tmp = x;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (1.0 - y) + (y * Math.exp(z));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = x - (Math.log1p((y * z)) / t);
	} else if (t_1 <= 1e+63) {
		tmp = x - (y * (Math.expm1(z) / t));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (1.0 - y) + (y * math.exp(z))
	tmp = 0
	if t_1 <= 0.0:
		tmp = x - (math.log1p((y * z)) / t)
	elif t_1 <= 1e+63:
		tmp = x - (y * (math.expm1(z) / t))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(1.0 - y) + Float64(y * exp(z)))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(x - Float64(log1p(Float64(y * z)) / t));
	elseif (t_1 <= 1e+63)
		tmp = Float64(x - Float64(y * Float64(expm1(z) / t)));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(1.0 - y), $MachinePrecision] + N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(x - N[(N[Log[1 + N[(y * z), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+63], N[(x - N[(y * N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 2.3%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 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))) < 1.00000000000000006e63

   1. Initial program 79.3%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 99.4

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

   5. Simplified 99.4%

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

   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. div-sub N/A

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

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

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

      4. div-sub N/A

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

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

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

      6. accelerator-lowering-expm1.f64 95.3

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

   8. Simplified 95.3%

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

   if 1.00000000000000006e63 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))

   1. Initial program 98.3%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 56.2%

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

Alternative 6: 88.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - y\right) + y \cdot e^{z} \leq 10^{+63}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ (- 1.0 y) (* y (exp z))) 1e+63) (- x (* y (/ (expm1 z) t))) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((1.0 - y) + (y * exp(z))) <= 1e+63) {
		tmp = x - (y * (expm1(z) / t));
	} else {
		tmp = x;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((1.0 - y) + (y * Math.exp(z))) <= 1e+63) {
		tmp = x - (y * (Math.expm1(z) / t));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((1.0 - y) + (y * math.exp(z))) <= 1e+63:
		tmp = x - (y * (math.expm1(z) / t))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(1.0 - y) + Float64(y * exp(z))) <= 1e+63)
		tmp = Float64(x - Float64(y * Float64(expm1(z) / t)));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(1.0 - y), $MachinePrecision] + N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+63], N[(x - N[(y * N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))) < 1.00000000000000006e63

   1. Initial program 55.6%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 99.5

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

   5. Simplified 99.5%

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

   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. div-sub N/A

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

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

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

      4. div-sub N/A

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

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

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

      6. accelerator-lowering-expm1.f64 89.6

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

   8. Simplified 89.6%

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

   if 1.00000000000000006e63 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))

   1. Initial program 98.3%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 56.2%

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

Alternative 7: 81.4% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{-69}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{fma}\left(z \cdot z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), z\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.65e-69)
   x
   (-
    x
    (*
     y
     (/
      (fma
       (* z z)
       (fma z (fma z 0.041666666666666664 0.16666666666666666) 0.5)
       z)
      t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.65e-69) {
		tmp = x;
	} else {
		tmp = x - (y * (fma((z * z), fma(z, fma(z, 0.041666666666666664, 0.16666666666666666), 0.5), z) / t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.65e-69)
		tmp = x;
	else
		tmp = Float64(x - Float64(y * Float64(fma(Float64(z * z), fma(z, fma(z, 0.041666666666666664, 0.16666666666666666), 0.5), z) / t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.65e-69], x, N[(x - N[(y * N[(N[(N[(z * z), $MachinePrecision] * N[(z * N[(z * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.65e-69

   1. Initial program 77.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 61.1%

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

   if -1.65e-69 < z

   1. Initial program 48.9%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 99.4

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

   5. Simplified 99.4%

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

   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. div-sub N/A

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

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

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

      4. div-sub N/A

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

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

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

      6. accelerator-lowering-expm1.f64 92.4

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

   8. Simplified 92.4%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

          \[\leadsto x - y \cdot \frac{\color{blue}{{z}^{2}} \cdot \left(\frac{1}{2} + z \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot z\right)\right) + z \cdot 1}{t} \]

       5. *-rgt-identity N/A

          \[\leadsto x - y \cdot \frac{{z}^{2} \cdot \left(\frac{1}{2} + z \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot z\right)\right) + \color{blue}{z}}{t} \]

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

          \[\leadsto x - y \cdot \frac{\color{blue}{\mathsf{fma}\left({z}^{2}, \frac{1}{2} + z \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot z\right), z\right)}}{t} \]

       7. unpow2 N/A

          \[\leadsto x - y \cdot \frac{\mathsf{fma}\left(\color{blue}{z \cdot z}, \frac{1}{2} + z \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot z\right), z\right)}{t} \]

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

          \[\leadsto x - y \cdot \frac{\mathsf{fma}\left(\color{blue}{z \cdot z}, \frac{1}{2} + z \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot z\right), z\right)}{t} \]

       9. +-commutative N/A

          \[\leadsto x - y \cdot \frac{\mathsf{fma}\left(z \cdot z, \color{blue}{z \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot z\right) + \frac{1}{2}}, z\right)}{t} \]

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

           \[\leadsto x - y \cdot \frac{\mathsf{fma}\left(z \cdot z, \color{blue}{\mathsf{fma}\left(z, \frac{1}{6} + \frac{1}{24} \cdot z, \frac{1}{2}\right)}, z\right)}{t} \]

       11. +-commutative N/A

           \[\leadsto x - y \cdot \frac{\mathsf{fma}\left(z \cdot z, \mathsf{fma}\left(z, \color{blue}{\frac{1}{24} \cdot z + \frac{1}{6}}, \frac{1}{2}\right), z\right)}{t} \]

       12. *-commutative N/A

           \[\leadsto x - y \cdot \frac{\mathsf{fma}\left(z \cdot z, \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{24}} + \frac{1}{6}, \frac{1}{2}\right), z\right)}{t} \]

       13. accelerator-lowering-fma.f64 91.8

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

   11. Simplified 91.8%

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

Alternative 8: 81.4% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.11 \cdot 10^{-69}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(z, z \cdot \mathsf{fma}\left(z, 0.16666666666666666, 0.5\right), z\right)}{-t}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.11e-69)
   x
   (fma y (/ (fma z (* z (fma z 0.16666666666666666 0.5)) z) (- t)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.11e-69) {
		tmp = x;
	} else {
		tmp = fma(y, (fma(z, (z * fma(z, 0.16666666666666666, 0.5)), z) / -t), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.11e-69)
		tmp = x;
	else
		tmp = fma(y, Float64(fma(z, Float64(z * fma(z, 0.16666666666666666, 0.5)), z) / Float64(-t)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.11e-69], x, N[(y * N[(N[(z * N[(z * N[(z * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] / (-t)), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.10999999999999999e-69

   1. Initial program 77.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 61.1%

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

   if -1.10999999999999999e-69 < z

   1. Initial program 48.9%

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

   3. Taylor expanded in z around 0

      \[\leadsto x - \frac{\color{blue}{z \cdot \left(y + z \cdot \left(\frac{1}{6} \cdot \left(z \cdot \left(y + \left(-3 \cdot {y}^{2} + 2 \cdot {y}^{3}\right)\right)\right) + \frac{1}{2} \cdot \left(y + -1 \cdot {y}^{2}\right)\right)\right)}}{t} \]
   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 69.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

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

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

   8. Simplified 91.8%

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

Alternative 9: 81.4% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{-69}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z \cdot \mathsf{fma}\left(z, -0.5, -1\right)}{t}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.65e-69) x (fma y (/ (* z (fma z -0.5 -1.0)) t) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.65e-69) {
		tmp = x;
	} else {
		tmp = fma(y, ((z * fma(z, -0.5, -1.0)) / t), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.65e-69)
		tmp = x;
	else
		tmp = fma(y, Float64(Float64(z * fma(z, -0.5, -1.0)) / t), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.65e-69], x, N[(y * N[(N[(z * N[(z * -0.5 + -1.0), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.65e-69

   1. Initial program 77.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 61.1%

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

   if -1.65e-69 < z

   1. Initial program 48.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

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

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

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

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

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

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

      6. exp-lowering-exp.f64 75.6

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

   5. Simplified 75.6%

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

   6. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. div-sub N/A

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

      3. associate-/l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 91.8

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

   8. Simplified 91.8%

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

Alternative 10: 70.3% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-260}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-167}:\\ \;\;\;\;\frac{y \cdot z}{-t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.4e-260) x (if (<= x 8.2e-167) (/ (* y z) (- t)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.4e-260) {
		tmp = x;
	} else if (x <= 8.2e-167) {
		tmp = (y * z) / -t;
	} else {
		tmp = x;
	}
	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 (x <= (-2.4d-260)) then
        tmp = x
    else if (x <= 8.2d-167) then
        tmp = (y * z) / -t
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.4e-260) {
		tmp = x;
	} else if (x <= 8.2e-167) {
		tmp = (y * z) / -t;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -2.4e-260:
		tmp = x
	elif x <= 8.2e-167:
		tmp = (y * z) / -t
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.4e-260)
		tmp = x;
	elseif (x <= 8.2e-167)
		tmp = Float64(Float64(y * z) / Float64(-t));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -2.4e-260)
		tmp = x;
	elseif (x <= 8.2e-167)
		tmp = (y * z) / -t;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.4e-260], x, If[LessEqual[x, 8.2e-167], N[(N[(y * z), $MachinePrecision] / (-t)), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4000000000000001e-260 or 8.20000000000000036e-167 < x

   1. Initial program 68.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 78.4%

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

   if -2.4000000000000001e-260 < x < 8.20000000000000036e-167

   1. Initial program 16.2%

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

   3. Taylor expanded in z around 0

      \[\leadsto x - \frac{\color{blue}{z \cdot \left(y + z \cdot \left(\frac{1}{6} \cdot \left(z \cdot \left(y + \left(-3 \cdot {y}^{2} + 2 \cdot {y}^{3}\right)\right)\right) + \frac{1}{2} \cdot \left(y + -1 \cdot {y}^{2}\right)\right)\right)}}{t} \]
   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 42.2%

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

   6. Taylor expanded in z around 0

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

   8. Simplified 55.1%

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

   9. Taylor expanded in x around 0

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

       1. associate-*l/ N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

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

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

       5. mul-1-neg N/A

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

       6. distribute-neg-frac2 N/A

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

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

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

       8. neg-lowering-neg.f64 47.4

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

   11. Simplified 47.4%

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

   12. Taylor expanded in z around 0

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

       1. associate-*r/ N/A

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

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

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

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

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

       6. mul-1-neg N/A

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

       7. neg-lowering-neg.f64 48.7

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

   14. Simplified 48.7%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-260}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-167}:\\ \;\;\;\;\frac{y \cdot z}{-t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.5% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-262}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-167}:\\ \;\;\;\;-z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.02e-262) x (if (<= x 3.2e-167) (- (* z (/ y t))) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.02e-262) {
		tmp = x;
	} else if (x <= 3.2e-167) {
		tmp = -(z * (y / t));
	} else {
		tmp = x;
	}
	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 (x <= (-1.02d-262)) then
        tmp = x
    else if (x <= 3.2d-167) then
        tmp = -(z * (y / t))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.02e-262) {
		tmp = x;
	} else if (x <= 3.2e-167) {
		tmp = -(z * (y / t));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.02e-262:
		tmp = x
	elif x <= 3.2e-167:
		tmp = -(z * (y / t))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.02e-262)
		tmp = x;
	elseif (x <= 3.2e-167)
		tmp = Float64(-Float64(z * Float64(y / t)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.02e-262)
		tmp = x;
	elseif (x <= 3.2e-167)
		tmp = -(z * (y / t));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.02e-262], x, If[LessEqual[x, 3.2e-167], (-N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.01999999999999994e-262 or 3.2000000000000002e-167 < x

   1. Initial program 68.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 78.4%

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

   if -1.01999999999999994e-262 < x < 3.2000000000000002e-167

   1. Initial program 16.2%

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

   3. Taylor expanded in z around 0

      \[\leadsto x - \frac{\color{blue}{z \cdot \left(y + z \cdot \left(\frac{1}{6} \cdot \left(z \cdot \left(y + \left(-3 \cdot {y}^{2} + 2 \cdot {y}^{3}\right)\right)\right) + \frac{1}{2} \cdot \left(y + -1 \cdot {y}^{2}\right)\right)\right)}}{t} \]
   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 42.2%

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

   6. Taylor expanded in z around 0

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

   8. Simplified 55.1%

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

   9. Taylor expanded in x around 0

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

       1. associate-*l/ N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

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

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

       5. mul-1-neg N/A

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

       6. distribute-neg-frac2 N/A

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

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

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

       8. neg-lowering-neg.f64 47.4

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

   11. Simplified 47.4%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-262}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-167}:\\ \;\;\;\;-z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.7% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-269}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-167}:\\ \;\;\;\;-y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -3.1e-269) x (if (<= x 3.5e-167) (- (* y (/ z t))) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3.1e-269) {
		tmp = x;
	} else if (x <= 3.5e-167) {
		tmp = -(y * (z / t));
	} else {
		tmp = x;
	}
	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 (x <= (-3.1d-269)) then
        tmp = x
    else if (x <= 3.5d-167) then
        tmp = -(y * (z / t))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3.1e-269) {
		tmp = x;
	} else if (x <= 3.5e-167) {
		tmp = -(y * (z / t));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -3.1e-269:
		tmp = x
	elif x <= 3.5e-167:
		tmp = -(y * (z / t))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -3.1e-269)
		tmp = x;
	elseif (x <= 3.5e-167)
		tmp = Float64(-Float64(y * Float64(z / t)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -3.1e-269)
		tmp = x;
	elseif (x <= 3.5e-167)
		tmp = -(y * (z / t));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -3.1e-269], x, If[LessEqual[x, 3.5e-167], (-N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), x]]

Derivation

1. Split input into 2 regimes

2. if x < -3.09999999999999967e-269 or 3.4999999999999999e-167 < x

   1. Initial program 67.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 77.4%

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

   if -3.09999999999999967e-269 < x < 3.4999999999999999e-167

   1. Initial program 17.4%

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

   3. Taylor expanded in z around 0

      \[\leadsto x - \frac{\color{blue}{z \cdot \left(y + z \cdot \left(\frac{1}{6} \cdot \left(z \cdot \left(y + \left(-3 \cdot {y}^{2} + 2 \cdot {y}^{3}\right)\right)\right) + \frac{1}{2} \cdot \left(y + -1 \cdot {y}^{2}\right)\right)\right)}}{t} \]
   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 45.7%

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

   6. Taylor expanded in z around 0

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

   8. Simplified 58.1%

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

   9. Taylor expanded in x around 0

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

       1. associate-*l/ N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

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

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

       5. mul-1-neg N/A

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

       6. distribute-neg-frac2 N/A

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

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

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

       8. neg-lowering-neg.f64 49.7

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

   11. Simplified 49.7%

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. associate-/l* N/A

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

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

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

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

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

       6. neg-lowering-neg.f64 50.8

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

   13. Applied egg-rr 50.8%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-269}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-167}:\\ \;\;\;\;-y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 81.3% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{-69}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -\frac{z}{t}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.65e-69) x (fma y (- (/ z t)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.65e-69) {
		tmp = x;
	} else {
		tmp = fma(y, -(z / t), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.65e-69)
		tmp = x;
	else
		tmp = fma(y, Float64(-Float64(z / t)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.65e-69], x, N[(y * (-N[(z / t), $MachinePrecision]) + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.65e-69

   1. Initial program 77.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 61.1%

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

   if -1.65e-69 < z

   1. Initial program 48.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

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

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

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

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

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

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

      6. exp-lowering-exp.f64 75.6

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

   5. Simplified 75.6%

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

   6. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

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

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

      3. /-lowering-/.f64 91.7

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

   8. Simplified 91.7%

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

Alternative 14: 80.5% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.62 \cdot 10^{-69}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.62e-69) x (- x (/ (* y z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.62e-69) {
		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 <= (-1.62d-69)) 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 <= -1.62e-69) {
		tmp = x;
	} else {
		tmp = x - ((y * z) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.62e-69:
		tmp = x
	else:
		tmp = x - ((y * z) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.62e-69)
		tmp = x;
	else
		tmp = Float64(x - Float64(Float64(y * z) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.62e-69)
		tmp = x;
	else
		tmp = x - ((y * z) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.62e-69], x, N[(x - N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.62e-69

   1. Initial program 77.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 61.1%

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

   if -1.62e-69 < z

   1. Initial program 48.9%

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

   3. Taylor expanded in z around 0

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

      1. *-lowering-*.f64 91.4

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

   5. Simplified 91.4%

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

Alternative 15: 71.4% 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 60.9%

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

3. Taylor expanded in x around inf

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

5. Simplified 69.0%

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

Developer Target 1: 74.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024198 
(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)))