Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A

Percentage Accurate: 98.4% → 98.3%

Time: 18.6s

Alternatives: 21

Speedup: 1.0×

• 47.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{e^{\mathsf{fma}\left(y, \log z, \left(t + -1\right) \cdot \log a - b\right) + \log x\_m}}{y} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (*
  x_s
  (/ (exp (+ (fma y (log z) (- (* (+ t -1.0) (log a)) b)) (log x_m))) y)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	return x_s * (exp((fma(y, log(z), (((t + -1.0) * log(a)) - b)) + log(x_m))) / y);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	return Float64(x_s * Float64(exp(Float64(fma(y, log(z), Float64(Float64(Float64(t + -1.0) * log(a)) - b)) + log(x_m))) / y))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * N[(N[Exp[N[(N[(y * N[Log[z], $MachinePrecision] + N[(N[(N[(t + -1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] + N[Log[x$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.3%

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

   1. add-exp-log 74.1%

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

   2. *-commutative 74.1%

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

   3. log-prod 50.2%

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

   4. add-log-exp 50.2%

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

   5. associate--l+ 50.2%

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

   6. fma-define 50.2%

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

   7. sub-neg 50.2%

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

   8. metadata-eval 50.2%

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

4. Applied egg-rr 50.2%

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

5. Final simplification 50.2%

   \[\leadsto \frac{e^{\mathsf{fma}\left(y, \log z, \left(t + -1\right) \cdot \log a - b\right) + \log x}}{y} \]
6. Add Preprocessing

Alternative 2: 89.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+149} \lor \neg \left(y \leq 8.2 \cdot 10^{+72}\right):\\ \;\;\;\;\frac{x\_m \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(\log x\_m + \left(t + -1\right) \cdot \log a\right) - b}}{y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (*
  x_s
  (if (or (<= y -6.5e+149) (not (<= y 8.2e+72)))
    (/ (* x_m (/ (pow z y) a)) y)
    (/ (exp (- (+ (log x_m) (* (+ t -1.0) (log a))) b)) y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -6.5e+149) || !(y <= 8.2e+72)) {
		tmp = (x_m * (pow(z, y) / a)) / y;
	} else {
		tmp = exp(((log(x_m) + ((t + -1.0) * log(a))) - b)) / y;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-6.5d+149)) .or. (.not. (y <= 8.2d+72))) then
        tmp = (x_m * ((z ** y) / a)) / y
    else
        tmp = exp(((log(x_m) + ((t + (-1.0d0)) * log(a))) - b)) / y
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -6.5e+149) || !(y <= 8.2e+72)) {
		tmp = (x_m * (Math.pow(z, y) / a)) / y;
	} else {
		tmp = Math.exp(((Math.log(x_m) + ((t + -1.0) * Math.log(a))) - b)) / y;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	tmp = 0
	if (y <= -6.5e+149) or not (y <= 8.2e+72):
		tmp = (x_m * (math.pow(z, y) / a)) / y
	else:
		tmp = math.exp(((math.log(x_m) + ((t + -1.0) * math.log(a))) - b)) / y
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -6.5e+149) || !(y <= 8.2e+72))
		tmp = Float64(Float64(x_m * Float64((z ^ y) / a)) / y);
	else
		tmp = Float64(exp(Float64(Float64(log(x_m) + Float64(Float64(t + -1.0) * log(a))) - b)) / y);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -6.5e+149) || ~((y <= 8.2e+72)))
		tmp = (x_m * ((z ^ y) / a)) / y;
	else
		tmp = exp(((log(x_m) + ((t + -1.0) * log(a))) - b)) / y;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * If[Or[LessEqual[y, -6.5e+149], N[Not[LessEqual[y, 8.2e+72]], $MachinePrecision]], N[(N[(x$95$m * N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[Exp[N[(N[(N[Log[x$95$m], $MachinePrecision] + N[(N[(t + -1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -6.50000000000000015e149 or 8.19999999999999926e72 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0 96.9%

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

      1. exp-sum 76.6%

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

      2. *-commutative 76.6%

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

      3. exp-to-pow 76.6%

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

      4. exp-to-pow 76.6%

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

      5. sub-neg 76.6%

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

      6. metadata-eval 76.6%

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

   5. Simplified 76.6%

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

   6. Taylor expanded in t around 0 93.8%

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

      1. associate-/l* 93.8%

         \[\leadsto \frac{\color{blue}{x \cdot \frac{{z}^{y}}{a}}}{y} \]

   8. Simplified 93.8%

      \[\leadsto \frac{\color{blue}{x \cdot \frac{{z}^{y}}{a}}}{y} \]

   if -6.50000000000000015e149 < y < 8.19999999999999926e72

   1. Initial program 97.7%

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

      1. add-exp-log 72.2%

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

      2. *-commutative 72.2%

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

      3. log-prod 53.4%

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

      4. add-log-exp 53.4%

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

      5. associate--l+ 53.4%

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

      6. fma-define 53.4%

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

      7. sub-neg 53.4%

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

      8. metadata-eval 53.4%

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

   4. Applied egg-rr 53.4%

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

   5. Taylor expanded in y around 0 50.1%

      \[\leadsto \frac{\color{blue}{e^{\left(\log x + \log a \cdot \left(t - 1\right)\right) - b}}}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+149} \lor \neg \left(y \leq 8.2 \cdot 10^{+72}\right):\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(\log x + \left(t + -1\right) \cdot \log a\right) - b}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+144} \lor \neg \left(y \leq 1.22\right):\\ \;\;\;\;\frac{x\_m \cdot e^{\left(y \cdot \log z - \log a\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(\log x\_m + \left(t + -1\right) \cdot \log a\right) - b}}{y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (*
  x_s
  (if (or (<= y -1.7e+144) (not (<= y 1.22)))
    (/ (* x_m (exp (- (- (* y (log z)) (log a)) b))) y)
    (/ (exp (- (+ (log x_m) (* (+ t -1.0) (log a))) b)) y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.7e+144) || !(y <= 1.22)) {
		tmp = (x_m * exp((((y * log(z)) - log(a)) - b))) / y;
	} else {
		tmp = exp(((log(x_m) + ((t + -1.0) * log(a))) - b)) / y;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-1.7d+144)) .or. (.not. (y <= 1.22d0))) then
        tmp = (x_m * exp((((y * log(z)) - log(a)) - b))) / y
    else
        tmp = exp(((log(x_m) + ((t + (-1.0d0)) * log(a))) - b)) / y
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.7e+144) || !(y <= 1.22)) {
		tmp = (x_m * Math.exp((((y * Math.log(z)) - Math.log(a)) - b))) / y;
	} else {
		tmp = Math.exp(((Math.log(x_m) + ((t + -1.0) * Math.log(a))) - b)) / y;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	tmp = 0
	if (y <= -1.7e+144) or not (y <= 1.22):
		tmp = (x_m * math.exp((((y * math.log(z)) - math.log(a)) - b))) / y
	else:
		tmp = math.exp(((math.log(x_m) + ((t + -1.0) * math.log(a))) - b)) / y
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.7e+144) || !(y <= 1.22))
		tmp = Float64(Float64(x_m * exp(Float64(Float64(Float64(y * log(z)) - log(a)) - b))) / y);
	else
		tmp = Float64(exp(Float64(Float64(log(x_m) + Float64(Float64(t + -1.0) * log(a))) - b)) / y);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.7e+144) || ~((y <= 1.22)))
		tmp = (x_m * exp((((y * log(z)) - log(a)) - b))) / y;
	else
		tmp = exp(((log(x_m) + ((t + -1.0) * log(a))) - b)) / y;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * If[Or[LessEqual[y, -1.7e+144], N[Not[LessEqual[y, 1.22]], $MachinePrecision]], N[(N[(x$95$m * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] - N[Log[a], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[Exp[N[(N[(N[Log[x$95$m], $MachinePrecision] + N[(N[(t + -1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -1.7e144 or 1.21999999999999997 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 97.3%

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

      1. +-commutative 97.3%

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

      2. mul-1-neg 97.3%

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

      3. unsub-neg 97.3%

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

   5. Simplified 97.3%

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

   if -1.7e144 < y < 1.21999999999999997

   1. Initial program 97.6%

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

      1. add-exp-log 71.4%

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

      2. *-commutative 71.4%

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

      3. log-prod 53.8%

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

      4. add-log-exp 53.8%

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

      5. associate--l+ 53.8%

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

      6. fma-define 53.8%

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

      7. sub-neg 53.8%

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

      8. metadata-eval 53.8%

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

   4. Applied egg-rr 53.8%

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

   5. Taylor expanded in y around 0 51.4%

      \[\leadsto \frac{\color{blue}{e^{\left(\log x + \log a \cdot \left(t - 1\right)\right) - b}}}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+144} \lor \neg \left(y \leq 1.22\right):\\ \;\;\;\;\frac{x \cdot e^{\left(y \cdot \log z - \log a\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(\log x + \left(t + -1\right) \cdot \log a\right) - b}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{x\_m \cdot e^{\left(y \cdot \log z + \left(t + -1\right) \cdot \log a\right) - b}}{y} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (* x_s (/ (* x_m (exp (- (+ (* y (log z)) (* (+ t -1.0) (log a))) b))) y)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	return x_s * ((x_m * exp((((y * log(z)) + ((t + -1.0) * log(a))) - b))) / y);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x_s * ((x_m * exp((((y * log(z)) + ((t + (-1.0d0)) * log(a))) - b))) / y)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	return x_s * ((x_m * Math.exp((((y * Math.log(z)) + ((t + -1.0) * Math.log(a))) - b))) / y);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	return x_s * ((x_m * math.exp((((y * math.log(z)) + ((t + -1.0) * math.log(a))) - b))) / y)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	return Float64(x_s * Float64(Float64(x_m * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t + -1.0) * log(a))) - b))) / y))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z, t, a, b)
	tmp = x_s * ((x_m * exp((((y * log(z)) + ((t + -1.0) * log(a))) - b))) / y);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * N[(N[(x$95$m * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t + -1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.3%

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

3. Final simplification 98.3%

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

Alternative 5: 89.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+146} \lor \neg \left(y \leq 2.3 \cdot 10^{+69}\right):\\ \;\;\;\;\frac{x\_m \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot e^{\left(t + -1\right) \cdot \log a - b}}{y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (*
  x_s
  (if (or (<= y -1.6e+146) (not (<= y 2.3e+69)))
    (/ (* x_m (/ (pow z y) a)) y)
    (/ (* x_m (exp (- (* (+ t -1.0) (log a)) b))) y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.6e+146) || !(y <= 2.3e+69)) {
		tmp = (x_m * (pow(z, y) / a)) / y;
	} else {
		tmp = (x_m * exp((((t + -1.0) * log(a)) - b))) / y;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-1.6d+146)) .or. (.not. (y <= 2.3d+69))) then
        tmp = (x_m * ((z ** y) / a)) / y
    else
        tmp = (x_m * exp((((t + (-1.0d0)) * log(a)) - b))) / y
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.6e+146) || !(y <= 2.3e+69)) {
		tmp = (x_m * (Math.pow(z, y) / a)) / y;
	} else {
		tmp = (x_m * Math.exp((((t + -1.0) * Math.log(a)) - b))) / y;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	tmp = 0
	if (y <= -1.6e+146) or not (y <= 2.3e+69):
		tmp = (x_m * (math.pow(z, y) / a)) / y
	else:
		tmp = (x_m * math.exp((((t + -1.0) * math.log(a)) - b))) / y
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.6e+146) || !(y <= 2.3e+69))
		tmp = Float64(Float64(x_m * Float64((z ^ y) / a)) / y);
	else
		tmp = Float64(Float64(x_m * exp(Float64(Float64(Float64(t + -1.0) * log(a)) - b))) / y);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.6e+146) || ~((y <= 2.3e+69)))
		tmp = (x_m * ((z ^ y) / a)) / y;
	else
		tmp = (x_m * exp((((t + -1.0) * log(a)) - b))) / y;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * If[Or[LessEqual[y, -1.6e+146], N[Not[LessEqual[y, 2.3e+69]], $MachinePrecision]], N[(N[(x$95$m * N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x$95$m * N[Exp[N[(N[(N[(t + -1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -1.6e146 or 2.30000000000000017e69 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0 96.9%

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

      1. exp-sum 76.6%

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

      2. *-commutative 76.6%

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

      3. exp-to-pow 76.6%

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

      4. exp-to-pow 76.6%

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

      5. sub-neg 76.6%

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

      6. metadata-eval 76.6%

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

   5. Simplified 76.6%

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

   6. Taylor expanded in t around 0 93.8%

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

      1. associate-/l* 93.8%

         \[\leadsto \frac{\color{blue}{x \cdot \frac{{z}^{y}}{a}}}{y} \]

   8. Simplified 93.8%

      \[\leadsto \frac{\color{blue}{x \cdot \frac{{z}^{y}}{a}}}{y} \]

   if -1.6e146 < y < 2.30000000000000017e69

   1. Initial program 97.7%

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

   3. Taylor expanded in y around 0 92.9%

      \[\leadsto \frac{x \cdot \color{blue}{e^{\log a \cdot \left(t - 1\right) - b}}}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+146} \lor \neg \left(y \leq 2.3 \cdot 10^{+69}\right):\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(t + -1\right) \cdot \log a - b}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{+127} \lor \neg \left(t \leq 0.0015\right):\\ \;\;\;\;\frac{x\_m \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{{z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (*
  x_s
  (if (or (<= t -7e+127) (not (<= t 0.0015)))
    (/ (* x_m (pow a (+ t -1.0))) y)
    (* x_m (/ (pow z y) (* a (* y (exp b))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -7e+127) || !(t <= 0.0015)) {
		tmp = (x_m * pow(a, (t + -1.0))) / y;
	} else {
		tmp = x_m * (pow(z, y) / (a * (y * exp(b))));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((t <= (-7d+127)) .or. (.not. (t <= 0.0015d0))) then
        tmp = (x_m * (a ** (t + (-1.0d0)))) / y
    else
        tmp = x_m * ((z ** y) / (a * (y * exp(b))))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -7e+127) || !(t <= 0.0015)) {
		tmp = (x_m * Math.pow(a, (t + -1.0))) / y;
	} else {
		tmp = x_m * (Math.pow(z, y) / (a * (y * Math.exp(b))));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	tmp = 0
	if (t <= -7e+127) or not (t <= 0.0015):
		tmp = (x_m * math.pow(a, (t + -1.0))) / y
	else:
		tmp = x_m * (math.pow(z, y) / (a * (y * math.exp(b))))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -7e+127) || !(t <= 0.0015))
		tmp = Float64(Float64(x_m * (a ^ Float64(t + -1.0))) / y);
	else
		tmp = Float64(x_m * Float64((z ^ y) / Float64(a * Float64(y * exp(b)))));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -7e+127) || ~((t <= 0.0015)))
		tmp = (x_m * (a ^ (t + -1.0))) / y;
	else
		tmp = x_m * ((z ^ y) / (a * (y * exp(b))));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * If[Or[LessEqual[t, -7e+127], N[Not[LessEqual[t, 0.0015]], $MachinePrecision]], N[(N[(x$95$m * N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(x$95$m * N[(N[Power[z, y], $MachinePrecision] / N[(a * N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -6.99999999999999956e127 or 0.0015 < t

   1. Initial program 99.0%

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

   3. Taylor expanded in b around 0 86.6%

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

      1. exp-sum 69.7%

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

      2. *-commutative 69.7%

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

      3. exp-to-pow 69.7%

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

      4. exp-to-pow 69.7%

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

      5. sub-neg 69.7%

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

      6. metadata-eval 69.7%

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

   5. Simplified 69.7%

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

   6. Taylor expanded in y around 0 81.4%

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

      1. exp-to-pow 81.4%

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

      2. sub-neg 81.4%

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

      3. metadata-eval 81.4%

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

   8. Simplified 81.4%

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

   if -6.99999999999999956e127 < t < 0.0015

   1. Initial program 97.9%

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

      1. associate-/l* 97.3%

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

      2. associate--l+ 97.3%

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

      3. exp-sum 86.8%

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

      4. associate-/l* 85.6%

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

      5. *-commutative 85.6%

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

      6. exp-to-pow 85.6%

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

      7. exp-diff 81.2%

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

      8. *-commutative 81.2%

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

      9. exp-to-pow 82.6%

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

      10. sub-neg 82.6%

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

      11. metadata-eval 82.6%

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

   3. Simplified 82.6%

      \[\leadsto \color{blue}{x \cdot \left({z}^{y} \cdot \frac{\frac{{a}^{\left(t + -1\right)}}{e^{b}}}{y}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 87.0%

      \[\leadsto x \cdot \color{blue}{\frac{{z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{+127} \lor \neg \left(t \leq 0.0015\right):\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -3.1 \cdot 10^{+128} \lor \neg \left(t \leq 3.7 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{x\_m \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (*
  x_s
  (if (or (<= t -3.1e+128) (not (<= t 3.7e-6)))
    (/ (* x_m (pow a (+ t -1.0))) y)
    (/ (* x_m (pow z y)) (* a (* y (exp b)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -3.1e+128) || !(t <= 3.7e-6)) {
		tmp = (x_m * pow(a, (t + -1.0))) / y;
	} else {
		tmp = (x_m * pow(z, y)) / (a * (y * exp(b)));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((t <= (-3.1d+128)) .or. (.not. (t <= 3.7d-6))) then
        tmp = (x_m * (a ** (t + (-1.0d0)))) / y
    else
        tmp = (x_m * (z ** y)) / (a * (y * exp(b)))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -3.1e+128) || !(t <= 3.7e-6)) {
		tmp = (x_m * Math.pow(a, (t + -1.0))) / y;
	} else {
		tmp = (x_m * Math.pow(z, y)) / (a * (y * Math.exp(b)));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	tmp = 0
	if (t <= -3.1e+128) or not (t <= 3.7e-6):
		tmp = (x_m * math.pow(a, (t + -1.0))) / y
	else:
		tmp = (x_m * math.pow(z, y)) / (a * (y * math.exp(b)))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -3.1e+128) || !(t <= 3.7e-6))
		tmp = Float64(Float64(x_m * (a ^ Float64(t + -1.0))) / y);
	else
		tmp = Float64(Float64(x_m * (z ^ y)) / Float64(a * Float64(y * exp(b))));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -3.1e+128) || ~((t <= 3.7e-6)))
		tmp = (x_m * (a ^ (t + -1.0))) / y;
	else
		tmp = (x_m * (z ^ y)) / (a * (y * exp(b)));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * If[Or[LessEqual[t, -3.1e+128], N[Not[LessEqual[t, 3.7e-6]], $MachinePrecision]], N[(N[(x$95$m * N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x$95$m * N[Power[z, y], $MachinePrecision]), $MachinePrecision] / N[(a * N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -3.10000000000000004e128 or 3.7000000000000002e-6 < t

   1. Initial program 99.0%

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

   3. Taylor expanded in b around 0 86.6%

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

      1. exp-sum 69.7%

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

      2. *-commutative 69.7%

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

      3. exp-to-pow 69.7%

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

      4. exp-to-pow 69.7%

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

      5. sub-neg 69.7%

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

      6. metadata-eval 69.7%

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

   5. Simplified 69.7%

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

   6. Taylor expanded in y around 0 81.4%

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

      1. exp-to-pow 81.4%

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

      2. sub-neg 81.4%

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

      3. metadata-eval 81.4%

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

   8. Simplified 81.4%

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

   if -3.10000000000000004e128 < t < 3.7000000000000002e-6

   1. Initial program 97.9%

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

      1. associate-/l* 97.3%

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

      2. associate--l+ 97.3%

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

      3. exp-sum 86.8%

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

      4. associate-/l* 85.6%

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

      5. *-commutative 85.6%

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

      6. exp-to-pow 85.6%

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

      7. exp-diff 81.2%

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

      8. *-commutative 81.2%

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

      9. exp-to-pow 82.6%

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

      10. sub-neg 82.6%

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

      11. metadata-eval 82.6%

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

   3. Simplified 82.6%

      \[\leadsto \color{blue}{x \cdot \left({z}^{y} \cdot \frac{\frac{{a}^{\left(t + -1\right)}}{e^{b}}}{y}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 87.3%

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.1 \cdot 10^{+128} \lor \neg \left(t \leq 3.7 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{x\_m \cdot {a}^{\left(t + -1\right)}}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+46}:\\ \;\;\;\;\frac{e^{-b}}{y}\\ \mathbf{elif}\;b \leq -2.4 \cdot 10^{-193}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -4.8 \cdot 10^{-264}:\\ \;\;\;\;\frac{x\_m \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{elif}\;b \leq 1.38 \cdot 10^{-191}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.38 \cdot 10^{-14}:\\ \;\;\;\;x\_m \cdot \frac{{z}^{y}}{y \cdot a}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{+26}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x_m (pow a (+ t -1.0))) y)))
   (*
    x_s
    (if (<= b -6e+46)
      (/ (exp (- b)) y)
      (if (<= b -2.4e-193)
        t_1
        (if (<= b -4.8e-264)
          (/ (* x_m (/ (pow z y) a)) y)
          (if (<= b 1.38e-191)
            t_1
            (if (<= b 1.38e-14)
              (* x_m (/ (pow z y) (* y a)))
              (if (<= b 3.6e+26) t_1 (/ x_m (* a (* y (exp b)))))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double t_1 = (x_m * pow(a, (t + -1.0))) / y;
	double tmp;
	if (b <= -6e+46) {
		tmp = exp(-b) / y;
	} else if (b <= -2.4e-193) {
		tmp = t_1;
	} else if (b <= -4.8e-264) {
		tmp = (x_m * (pow(z, y) / a)) / y;
	} else if (b <= 1.38e-191) {
		tmp = t_1;
	} else if (b <= 1.38e-14) {
		tmp = x_m * (pow(z, y) / (y * a));
	} else if (b <= 3.6e+26) {
		tmp = t_1;
	} else {
		tmp = x_m / (a * (y * exp(b)));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x_m * (a ** (t + (-1.0d0)))) / y
    if (b <= (-6d+46)) then
        tmp = exp(-b) / y
    else if (b <= (-2.4d-193)) then
        tmp = t_1
    else if (b <= (-4.8d-264)) then
        tmp = (x_m * ((z ** y) / a)) / y
    else if (b <= 1.38d-191) then
        tmp = t_1
    else if (b <= 1.38d-14) then
        tmp = x_m * ((z ** y) / (y * a))
    else if (b <= 3.6d+26) then
        tmp = t_1
    else
        tmp = x_m / (a * (y * exp(b)))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double t_1 = (x_m * Math.pow(a, (t + -1.0))) / y;
	double tmp;
	if (b <= -6e+46) {
		tmp = Math.exp(-b) / y;
	} else if (b <= -2.4e-193) {
		tmp = t_1;
	} else if (b <= -4.8e-264) {
		tmp = (x_m * (Math.pow(z, y) / a)) / y;
	} else if (b <= 1.38e-191) {
		tmp = t_1;
	} else if (b <= 1.38e-14) {
		tmp = x_m * (Math.pow(z, y) / (y * a));
	} else if (b <= 3.6e+26) {
		tmp = t_1;
	} else {
		tmp = x_m / (a * (y * Math.exp(b)));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	t_1 = (x_m * math.pow(a, (t + -1.0))) / y
	tmp = 0
	if b <= -6e+46:
		tmp = math.exp(-b) / y
	elif b <= -2.4e-193:
		tmp = t_1
	elif b <= -4.8e-264:
		tmp = (x_m * (math.pow(z, y) / a)) / y
	elif b <= 1.38e-191:
		tmp = t_1
	elif b <= 1.38e-14:
		tmp = x_m * (math.pow(z, y) / (y * a))
	elif b <= 3.6e+26:
		tmp = t_1
	else:
		tmp = x_m / (a * (y * math.exp(b)))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	t_1 = Float64(Float64(x_m * (a ^ Float64(t + -1.0))) / y)
	tmp = 0.0
	if (b <= -6e+46)
		tmp = Float64(exp(Float64(-b)) / y);
	elseif (b <= -2.4e-193)
		tmp = t_1;
	elseif (b <= -4.8e-264)
		tmp = Float64(Float64(x_m * Float64((z ^ y) / a)) / y);
	elseif (b <= 1.38e-191)
		tmp = t_1;
	elseif (b <= 1.38e-14)
		tmp = Float64(x_m * Float64((z ^ y) / Float64(y * a)));
	elseif (b <= 3.6e+26)
		tmp = t_1;
	else
		tmp = Float64(x_m / Float64(a * Float64(y * exp(b))));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t, a, b)
	t_1 = (x_m * (a ^ (t + -1.0))) / y;
	tmp = 0.0;
	if (b <= -6e+46)
		tmp = exp(-b) / y;
	elseif (b <= -2.4e-193)
		tmp = t_1;
	elseif (b <= -4.8e-264)
		tmp = (x_m * ((z ^ y) / a)) / y;
	elseif (b <= 1.38e-191)
		tmp = t_1;
	elseif (b <= 1.38e-14)
		tmp = x_m * ((z ^ y) / (y * a));
	elseif (b <= 3.6e+26)
		tmp = t_1;
	else
		tmp = x_m / (a * (y * exp(b)));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x$95$m * N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, N[(x$95$s * If[LessEqual[b, -6e+46], N[(N[Exp[(-b)], $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, -2.4e-193], t$95$1, If[LessEqual[b, -4.8e-264], N[(N[(x$95$m * N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, 1.38e-191], t$95$1, If[LessEqual[b, 1.38e-14], N[(x$95$m * N[(N[Power[z, y], $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.6e+26], t$95$1, N[(x$95$m / N[(a * N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]), $MachinePrecision]]

Derivation

1. Split input into 5 regimes

2. if b < -6.00000000000000047e46

   1. Initial program 100.0%

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

      1. add-exp-log 56.1%

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

      2. *-commutative 56.1%

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

      3. log-prod 53.7%

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

      4. add-log-exp 53.7%

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

      5. associate--l+ 53.7%

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

      6. fma-define 53.7%

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

      7. sub-neg 53.7%

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

      8. metadata-eval 53.7%

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

   4. Applied egg-rr 53.7%

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

   5. Taylor expanded in b around inf 51.3%

      \[\leadsto \frac{e^{\color{blue}{-1 \cdot b}}}{y} \]
   6. Step-by-step derivation

      1. neg-mul-1 51.3%

         \[\leadsto \frac{e^{\color{blue}{-b}}}{y} \]

   7. Simplified 51.3%

      \[\leadsto \frac{e^{\color{blue}{-b}}}{y} \]

   if -6.00000000000000047e46 < b < -2.4e-193 or -4.7999999999999997e-264 < b < 1.38000000000000003e-191 or 1.38000000000000002e-14 < b < 3.60000000000000024e26

   1. Initial program 96.0%

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

   3. Taylor expanded in b around 0 95.1%

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

      1. exp-sum 84.4%

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

      2. *-commutative 84.4%

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

      3. exp-to-pow 84.4%

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

      4. exp-to-pow 86.3%

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

      5. sub-neg 86.3%

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

      6. metadata-eval 86.3%

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

   5. Simplified 86.3%

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

   6. Taylor expanded in y around 0 83.9%

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

      1. exp-to-pow 85.7%

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

      2. sub-neg 85.7%

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

      3. metadata-eval 85.7%

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

   8. Simplified 85.7%

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

   if -2.4e-193 < b < -4.7999999999999997e-264

   1. Initial program 99.4%

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

   3. Taylor expanded in b around 0 99.4%

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

      1. exp-sum 77.9%

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

      2. *-commutative 77.9%

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

      3. exp-to-pow 77.9%

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

      4. exp-to-pow 78.5%

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

      5. sub-neg 78.5%

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

      6. metadata-eval 78.5%

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

   5. Simplified 78.5%

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

   6. Taylor expanded in t around 0 99.9%

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

      1. associate-/l* 99.9%

         \[\leadsto \frac{\color{blue}{x \cdot \frac{{z}^{y}}{a}}}{y} \]

   8. Simplified 99.9%

      \[\leadsto \frac{\color{blue}{x \cdot \frac{{z}^{y}}{a}}}{y} \]

   if 1.38000000000000003e-191 < b < 1.38000000000000002e-14

   1. Initial program 99.5%

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

      1. associate-/l* 99.5%

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

      2. associate--l+ 99.5%

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

      3. exp-sum 92.0%

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

      4. associate-/l* 92.0%

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

      5. *-commutative 92.0%

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

      6. exp-to-pow 92.0%

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

      7. exp-diff 92.0%

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

      8. *-commutative 92.0%

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

      9. exp-to-pow 92.5%

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

      10. sub-neg 92.5%

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

      11. metadata-eval 92.5%

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

   3. Simplified 92.5%

      \[\leadsto \color{blue}{x \cdot \left({z}^{y} \cdot \frac{\frac{{a}^{\left(t + -1\right)}}{e^{b}}}{y}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 78.0%

      \[\leadsto x \cdot \color{blue}{\frac{{z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]

   6. Taylor expanded in b around 0 78.0%

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

      1. associate-/l* 78.0%

         \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{a \cdot y}} \]

   8. Simplified 78.0%

      \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{a \cdot y}} \]

   if 3.60000000000000024e26 < b

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate--l+ 100.0%

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

      3. exp-sum 81.4%

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

      4. associate-/l* 81.4%

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

      5. *-commutative 81.4%

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

      6. exp-to-pow 81.4%

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

      7. exp-diff 61.0%

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

      8. *-commutative 61.0%

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

      9. exp-to-pow 61.0%

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

      10. sub-neg 61.0%

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

      11. metadata-eval 61.0%

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

   3. Simplified 61.0%

      \[\leadsto \color{blue}{x \cdot \left({z}^{y} \cdot \frac{\frac{{a}^{\left(t + -1\right)}}{e^{b}}}{y}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 78.0%

      \[\leadsto x \cdot \color{blue}{\frac{{z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]

   6. Taylor expanded in y around 0 86.7%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+46}:\\ \;\;\;\;\frac{e^{-b}}{y}\\ \mathbf{elif}\;b \leq -2.4 \cdot 10^{-193}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{elif}\;b \leq -4.8 \cdot 10^{-264}:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{elif}\;b \leq 1.38 \cdot 10^{-191}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{elif}\;b \leq 1.38 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{y \cdot a}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{+26}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot {a}^{\left(t + -1\right)}\\ t_2 := \frac{t\_1}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{+48}:\\ \;\;\;\;\frac{e^{-b}}{y}\\ \mathbf{elif}\;b \leq -2.85 \cdot 10^{-195}:\\ \;\;\;\;\frac{t\_1 \cdot \left(1 - b\right)}{y}\\ \mathbf{elif}\;b \leq -1.05 \cdot 10^{-262}:\\ \;\;\;\;\frac{x\_m \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-193}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;b \leq 1.28 \cdot 10^{-11}:\\ \;\;\;\;x\_m \cdot \frac{{z}^{y}}{y \cdot a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+25}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (let* ((t_1 (* x_m (pow a (+ t -1.0)))) (t_2 (/ t_1 y)))
   (*
    x_s
    (if (<= b -4.8e+48)
      (/ (exp (- b)) y)
      (if (<= b -2.85e-195)
        (/ (* t_1 (- 1.0 b)) y)
        (if (<= b -1.05e-262)
          (/ (* x_m (/ (pow z y) a)) y)
          (if (<= b 9e-193)
            t_2
            (if (<= b 1.28e-11)
              (* x_m (/ (pow z y) (* y a)))
              (if (<= b 2e+25) t_2 (/ x_m (* a (* y (exp b)))))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double t_1 = x_m * pow(a, (t + -1.0));
	double t_2 = t_1 / y;
	double tmp;
	if (b <= -4.8e+48) {
		tmp = exp(-b) / y;
	} else if (b <= -2.85e-195) {
		tmp = (t_1 * (1.0 - b)) / y;
	} else if (b <= -1.05e-262) {
		tmp = (x_m * (pow(z, y) / a)) / y;
	} else if (b <= 9e-193) {
		tmp = t_2;
	} else if (b <= 1.28e-11) {
		tmp = x_m * (pow(z, y) / (y * a));
	} else if (b <= 2e+25) {
		tmp = t_2;
	} else {
		tmp = x_m / (a * (y * exp(b)));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x_m * (a ** (t + (-1.0d0)))
    t_2 = t_1 / y
    if (b <= (-4.8d+48)) then
        tmp = exp(-b) / y
    else if (b <= (-2.85d-195)) then
        tmp = (t_1 * (1.0d0 - b)) / y
    else if (b <= (-1.05d-262)) then
        tmp = (x_m * ((z ** y) / a)) / y
    else if (b <= 9d-193) then
        tmp = t_2
    else if (b <= 1.28d-11) then
        tmp = x_m * ((z ** y) / (y * a))
    else if (b <= 2d+25) then
        tmp = t_2
    else
        tmp = x_m / (a * (y * exp(b)))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double t_1 = x_m * Math.pow(a, (t + -1.0));
	double t_2 = t_1 / y;
	double tmp;
	if (b <= -4.8e+48) {
		tmp = Math.exp(-b) / y;
	} else if (b <= -2.85e-195) {
		tmp = (t_1 * (1.0 - b)) / y;
	} else if (b <= -1.05e-262) {
		tmp = (x_m * (Math.pow(z, y) / a)) / y;
	} else if (b <= 9e-193) {
		tmp = t_2;
	} else if (b <= 1.28e-11) {
		tmp = x_m * (Math.pow(z, y) / (y * a));
	} else if (b <= 2e+25) {
		tmp = t_2;
	} else {
		tmp = x_m / (a * (y * Math.exp(b)));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	t_1 = x_m * math.pow(a, (t + -1.0))
	t_2 = t_1 / y
	tmp = 0
	if b <= -4.8e+48:
		tmp = math.exp(-b) / y
	elif b <= -2.85e-195:
		tmp = (t_1 * (1.0 - b)) / y
	elif b <= -1.05e-262:
		tmp = (x_m * (math.pow(z, y) / a)) / y
	elif b <= 9e-193:
		tmp = t_2
	elif b <= 1.28e-11:
		tmp = x_m * (math.pow(z, y) / (y * a))
	elif b <= 2e+25:
		tmp = t_2
	else:
		tmp = x_m / (a * (y * math.exp(b)))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	t_1 = Float64(x_m * (a ^ Float64(t + -1.0)))
	t_2 = Float64(t_1 / y)
	tmp = 0.0
	if (b <= -4.8e+48)
		tmp = Float64(exp(Float64(-b)) / y);
	elseif (b <= -2.85e-195)
		tmp = Float64(Float64(t_1 * Float64(1.0 - b)) / y);
	elseif (b <= -1.05e-262)
		tmp = Float64(Float64(x_m * Float64((z ^ y) / a)) / y);
	elseif (b <= 9e-193)
		tmp = t_2;
	elseif (b <= 1.28e-11)
		tmp = Float64(x_m * Float64((z ^ y) / Float64(y * a)));
	elseif (b <= 2e+25)
		tmp = t_2;
	else
		tmp = Float64(x_m / Float64(a * Float64(y * exp(b))));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t, a, b)
	t_1 = x_m * (a ^ (t + -1.0));
	t_2 = t_1 / y;
	tmp = 0.0;
	if (b <= -4.8e+48)
		tmp = exp(-b) / y;
	elseif (b <= -2.85e-195)
		tmp = (t_1 * (1.0 - b)) / y;
	elseif (b <= -1.05e-262)
		tmp = (x_m * ((z ^ y) / a)) / y;
	elseif (b <= 9e-193)
		tmp = t_2;
	elseif (b <= 1.28e-11)
		tmp = x_m * ((z ^ y) / (y * a));
	elseif (b <= 2e+25)
		tmp = t_2;
	else
		tmp = x_m / (a * (y * exp(b)));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x$95$m * N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / y), $MachinePrecision]}, N[(x$95$s * If[LessEqual[b, -4.8e+48], N[(N[Exp[(-b)], $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, -2.85e-195], N[(N[(t$95$1 * N[(1.0 - b), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, -1.05e-262], N[(N[(x$95$m * N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, 9e-193], t$95$2, If[LessEqual[b, 1.28e-11], N[(x$95$m * N[(N[Power[z, y], $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e+25], t$95$2, N[(x$95$m / N[(a * N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 6 regimes

2. if b < -4.8000000000000002e48

   1. Initial program 100.0%

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

      1. add-exp-log 56.1%

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

      2. *-commutative 56.1%

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

      3. log-prod 53.7%

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

      4. add-log-exp 53.7%

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

      5. associate--l+ 53.7%

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

      6. fma-define 53.7%

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

      7. sub-neg 53.7%

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

      8. metadata-eval 53.7%

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

   4. Applied egg-rr 53.7%

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

   5. Taylor expanded in b around inf 51.3%

      \[\leadsto \frac{e^{\color{blue}{-1 \cdot b}}}{y} \]
   6. Step-by-step derivation

      1. neg-mul-1 51.3%

         \[\leadsto \frac{e^{\color{blue}{-b}}}{y} \]

   7. Simplified 51.3%

      \[\leadsto \frac{e^{\color{blue}{-b}}}{y} \]

   if -4.8000000000000002e48 < b < -2.85e-195

   1. Initial program 94.5%

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

      1. add-exp-log 74.8%

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

      2. *-commutative 74.8%

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

      3. log-prod 51.0%

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

      4. add-log-exp 51.0%

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

      5. associate--l+ 51.0%

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

      6. fma-define 51.0%

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

      7. sub-neg 51.0%

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

      8. metadata-eval 51.0%

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

   4. Applied egg-rr 51.0%

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

   5. Taylor expanded in y around 0 50.1%

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

   6. Taylor expanded in b around 0 50.1%

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

      1. associate-*r* 50.1%

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

      2. neg-mul-1 50.1%

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

      3. distribute-rgt1-in 50.1%

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

      4. exp-sum 50.3%

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

      5. rem-exp-log 84.7%

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

      6. exp-to-pow 86.6%

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

      7. sub-neg 86.6%

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

      8. metadata-eval 86.6%

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

   8. Simplified 86.6%

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

   if -2.85e-195 < b < -1.05e-262

   1. Initial program 99.4%

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

   3. Taylor expanded in b around 0 99.4%

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

      1. exp-sum 77.9%

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

      2. *-commutative 77.9%

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

      3. exp-to-pow 77.9%

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

      4. exp-to-pow 78.5%

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

      5. sub-neg 78.5%

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

      6. metadata-eval 78.5%

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

   5. Simplified 78.5%

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

   6. Taylor expanded in t around 0 99.9%

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

      1. associate-/l* 99.9%

         \[\leadsto \frac{\color{blue}{x \cdot \frac{{z}^{y}}{a}}}{y} \]

   8. Simplified 99.9%

      \[\leadsto \frac{\color{blue}{x \cdot \frac{{z}^{y}}{a}}}{y} \]

   if -1.05e-262 < b < 8.9999999999999997e-193 or 1.28e-11 < b < 2.00000000000000018e25

   1. Initial program 97.8%

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

   3. Taylor expanded in b around 0 95.7%

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

      1. exp-sum 81.0%

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

      2. *-commutative 81.0%

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

      3. exp-to-pow 81.0%

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

      4. exp-to-pow 82.9%

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

      5. sub-neg 82.9%

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

      6. metadata-eval 82.9%

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

   5. Simplified 82.9%

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

   6. Taylor expanded in y around 0 83.0%

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

      1. exp-to-pow 84.8%

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

      2. sub-neg 84.8%

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

      3. metadata-eval 84.8%

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

   8. Simplified 84.8%

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

   if 8.9999999999999997e-193 < b < 1.28e-11

   1. Initial program 99.5%

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

      1. associate-/l* 99.5%

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

      2. associate--l+ 99.5%

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

      3. exp-sum 92.0%

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

      4. associate-/l* 92.0%

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

      5. *-commutative 92.0%

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

      6. exp-to-pow 92.0%

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

      7. exp-diff 92.0%

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

      8. *-commutative 92.0%

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

      9. exp-to-pow 92.5%

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

      10. sub-neg 92.5%

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

      11. metadata-eval 92.5%

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

   3. Simplified 92.5%

      \[\leadsto \color{blue}{x \cdot \left({z}^{y} \cdot \frac{\frac{{a}^{\left(t + -1\right)}}{e^{b}}}{y}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 78.0%

      \[\leadsto x \cdot \color{blue}{\frac{{z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]

   6. Taylor expanded in b around 0 78.0%

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

      1. associate-/l* 78.0%

         \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{a \cdot y}} \]

   8. Simplified 78.0%

      \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{a \cdot y}} \]

   if 2.00000000000000018e25 < b

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate--l+ 100.0%

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

      3. exp-sum 81.4%

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

      4. associate-/l* 81.4%

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

      5. *-commutative 81.4%

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

      6. exp-to-pow 81.4%

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

      7. exp-diff 61.0%

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

      8. *-commutative 61.0%

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

      9. exp-to-pow 61.0%

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

      10. sub-neg 61.0%

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

      11. metadata-eval 61.0%

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

   3. Simplified 61.0%

      \[\leadsto \color{blue}{x \cdot \left({z}^{y} \cdot \frac{\frac{{a}^{\left(t + -1\right)}}{e^{b}}}{y}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 78.0%

      \[\leadsto x \cdot \color{blue}{\frac{{z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]

   6. Taylor expanded in y around 0 86.7%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{+48}:\\ \;\;\;\;\frac{e^{-b}}{y}\\ \mathbf{elif}\;b \leq -2.85 \cdot 10^{-195}:\\ \;\;\;\;\frac{\left(x \cdot {a}^{\left(t + -1\right)}\right) \cdot \left(1 - b\right)}{y}\\ \mathbf{elif}\;b \leq -1.05 \cdot 10^{-262}:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-193}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{elif}\;b \leq 1.28 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{y \cdot a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+25}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{+46}:\\ \;\;\;\;\frac{e^{-b}}{y}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{+25}:\\ \;\;\;\;x\_m \cdot \frac{{z}^{y}}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (*
  x_s
  (if (<= b -7e+46)
    (/ (exp (- b)) y)
    (if (<= b 2.7e+25)
      (* x_m (/ (pow z y) (* y a)))
      (/ x_m (* a (* y (exp b))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -7e+46) {
		tmp = exp(-b) / y;
	} else if (b <= 2.7e+25) {
		tmp = x_m * (pow(z, y) / (y * a));
	} else {
		tmp = x_m / (a * (y * exp(b)));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-7d+46)) then
        tmp = exp(-b) / y
    else if (b <= 2.7d+25) then
        tmp = x_m * ((z ** y) / (y * a))
    else
        tmp = x_m / (a * (y * exp(b)))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -7e+46) {
		tmp = Math.exp(-b) / y;
	} else if (b <= 2.7e+25) {
		tmp = x_m * (Math.pow(z, y) / (y * a));
	} else {
		tmp = x_m / (a * (y * Math.exp(b)));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	tmp = 0
	if b <= -7e+46:
		tmp = math.exp(-b) / y
	elif b <= 2.7e+25:
		tmp = x_m * (math.pow(z, y) / (y * a))
	else:
		tmp = x_m / (a * (y * math.exp(b)))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0
	if (b <= -7e+46)
		tmp = Float64(exp(Float64(-b)) / y);
	elseif (b <= 2.7e+25)
		tmp = Float64(x_m * Float64((z ^ y) / Float64(y * a)));
	else
		tmp = Float64(x_m / Float64(a * Float64(y * exp(b))));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -7e+46)
		tmp = exp(-b) / y;
	elseif (b <= 2.7e+25)
		tmp = x_m * ((z ^ y) / (y * a));
	else
		tmp = x_m / (a * (y * exp(b)));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * If[LessEqual[b, -7e+46], N[(N[Exp[(-b)], $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, 2.7e+25], N[(x$95$m * N[(N[Power[z, y], $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(a * N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if b < -6.9999999999999997e46

   1. Initial program 100.0%

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

      1. add-exp-log 56.1%

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

      2. *-commutative 56.1%

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

      3. log-prod 53.7%

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

      4. add-log-exp 53.7%

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

      5. associate--l+ 53.7%

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

      6. fma-define 53.7%

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

      7. sub-neg 53.7%

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

      8. metadata-eval 53.7%

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

   4. Applied egg-rr 53.7%

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

   5. Taylor expanded in b around inf 51.3%

      \[\leadsto \frac{e^{\color{blue}{-1 \cdot b}}}{y} \]
   6. Step-by-step derivation

      1. neg-mul-1 51.3%

         \[\leadsto \frac{e^{\color{blue}{-b}}}{y} \]

   7. Simplified 51.3%

      \[\leadsto \frac{e^{\color{blue}{-b}}}{y} \]

   if -6.9999999999999997e46 < b < 2.7e25

   1. Initial program 97.2%

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

      1. associate-/l* 96.7%

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

      2. associate--l+ 96.7%

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

      3. exp-sum 84.5%

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

      4. associate-/l* 83.3%

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

      5. *-commutative 83.3%

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

      6. exp-to-pow 83.3%

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

      7. exp-diff 79.4%

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

      8. *-commutative 79.4%

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

      9. exp-to-pow 80.8%

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

      10. sub-neg 80.8%

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

      11. metadata-eval 80.8%

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

   3. Simplified 80.8%

      \[\leadsto \color{blue}{x \cdot \left({z}^{y} \cdot \frac{\frac{{a}^{\left(t + -1\right)}}{e^{b}}}{y}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 65.4%

      \[\leadsto x \cdot \color{blue}{\frac{{z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]

   6. Taylor expanded in b around 0 68.3%

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

      1. associate-/l* 68.0%

         \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{a \cdot y}} \]

   8. Simplified 68.0%

      \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{a \cdot y}} \]

   if 2.7e25 < b

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate--l+ 100.0%

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

      3. exp-sum 81.4%

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

      4. associate-/l* 81.4%

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

      5. *-commutative 81.4%

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

      6. exp-to-pow 81.4%

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

      7. exp-diff 61.0%

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

      8. *-commutative 61.0%

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

      9. exp-to-pow 61.0%

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

      10. sub-neg 61.0%

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

      11. metadata-eval 61.0%

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

   3. Simplified 61.0%

      \[\leadsto \color{blue}{x \cdot \left({z}^{y} \cdot \frac{\frac{{a}^{\left(t + -1\right)}}{e^{b}}}{y}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 78.0%

      \[\leadsto x \cdot \color{blue}{\frac{{z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]

   6. Taylor expanded in y around 0 86.7%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{+46}:\\ \;\;\;\;\frac{e^{-b}}{y}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{+25}:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq -2.15 \cdot 10^{+72}:\\ \;\;\;\;\frac{e^{-b}}{y}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{+25}:\\ \;\;\;\;\frac{x\_m \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (*
  x_s
  (if (<= b -2.15e+72)
    (/ (exp (- b)) y)
    (if (<= b 1.9e+25)
      (/ (* x_m (/ (pow z y) a)) y)
      (/ x_m (* a (* y (exp b))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -2.15e+72) {
		tmp = exp(-b) / y;
	} else if (b <= 1.9e+25) {
		tmp = (x_m * (pow(z, y) / a)) / y;
	} else {
		tmp = x_m / (a * (y * exp(b)));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-2.15d+72)) then
        tmp = exp(-b) / y
    else if (b <= 1.9d+25) then
        tmp = (x_m * ((z ** y) / a)) / y
    else
        tmp = x_m / (a * (y * exp(b)))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -2.15e+72) {
		tmp = Math.exp(-b) / y;
	} else if (b <= 1.9e+25) {
		tmp = (x_m * (Math.pow(z, y) / a)) / y;
	} else {
		tmp = x_m / (a * (y * Math.exp(b)));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	tmp = 0
	if b <= -2.15e+72:
		tmp = math.exp(-b) / y
	elif b <= 1.9e+25:
		tmp = (x_m * (math.pow(z, y) / a)) / y
	else:
		tmp = x_m / (a * (y * math.exp(b)))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0
	if (b <= -2.15e+72)
		tmp = Float64(exp(Float64(-b)) / y);
	elseif (b <= 1.9e+25)
		tmp = Float64(Float64(x_m * Float64((z ^ y) / a)) / y);
	else
		tmp = Float64(x_m / Float64(a * Float64(y * exp(b))));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -2.15e+72)
		tmp = exp(-b) / y;
	elseif (b <= 1.9e+25)
		tmp = (x_m * ((z ^ y) / a)) / y;
	else
		tmp = x_m / (a * (y * exp(b)));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * If[LessEqual[b, -2.15e+72], N[(N[Exp[(-b)], $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, 1.9e+25], N[(N[(x$95$m * N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(x$95$m / N[(a * N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if b < -2.1500000000000001e72

   1. Initial program 100.0%

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

      1. add-exp-log 57.9%

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

      2. *-commutative 57.9%

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

      3. log-prod 55.3%

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

      4. add-log-exp 55.3%

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

      5. associate--l+ 55.3%

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

      6. fma-define 55.3%

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

      7. sub-neg 55.3%

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

      8. metadata-eval 55.3%

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

   4. Applied egg-rr 55.3%

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

   5. Taylor expanded in b around inf 52.7%

      \[\leadsto \frac{e^{\color{blue}{-1 \cdot b}}}{y} \]
   6. Step-by-step derivation

      1. neg-mul-1 52.7%

         \[\leadsto \frac{e^{\color{blue}{-b}}}{y} \]

   7. Simplified 52.7%

      \[\leadsto \frac{e^{\color{blue}{-b}}}{y} \]

   if -2.1500000000000001e72 < b < 1.9e25

   1. Initial program 97.3%

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

   3. Taylor expanded in b around 0 96.6%

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

      1. exp-sum 85.4%

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

      2. *-commutative 85.4%

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

      3. exp-to-pow 85.4%

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

      4. exp-to-pow 86.8%

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

      5. sub-neg 86.8%

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

      6. metadata-eval 86.8%

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

   5. Simplified 86.8%

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

   6. Taylor expanded in t around 0 69.9%

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

      1. associate-/l* 69.9%

         \[\leadsto \frac{\color{blue}{x \cdot \frac{{z}^{y}}{a}}}{y} \]

   8. Simplified 69.9%

      \[\leadsto \frac{\color{blue}{x \cdot \frac{{z}^{y}}{a}}}{y} \]

   if 1.9e25 < b

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate--l+ 100.0%

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

      3. exp-sum 81.4%

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

      4. associate-/l* 81.4%

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

      5. *-commutative 81.4%

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

      6. exp-to-pow 81.4%

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

      7. exp-diff 61.0%

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

      8. *-commutative 61.0%

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

      9. exp-to-pow 61.0%

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

      10. sub-neg 61.0%

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

      11. metadata-eval 61.0%

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

   3. Simplified 61.0%

      \[\leadsto \color{blue}{x \cdot \left({z}^{y} \cdot \frac{\frac{{a}^{\left(t + -1\right)}}{e^{b}}}{y}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 78.0%

      \[\leadsto x \cdot \color{blue}{\frac{{z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]

   6. Taylor expanded in y around 0 86.7%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.15 \cdot 10^{+72}:\\ \;\;\;\;\frac{e^{-b}}{y}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{+25}:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{+46} \lor \neg \left(b \leq 660\right):\\ \;\;\;\;\frac{e^{-b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y \cdot a}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (*
  x_s
  (if (or (<= b -5.2e+46) (not (<= b 660.0)))
    (/ (exp (- b)) y)
    (/ x_m (* y a)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -5.2e+46) || !(b <= 660.0)) {
		tmp = exp(-b) / y;
	} else {
		tmp = x_m / (y * a);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-5.2d+46)) .or. (.not. (b <= 660.0d0))) then
        tmp = exp(-b) / y
    else
        tmp = x_m / (y * a)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -5.2e+46) || !(b <= 660.0)) {
		tmp = Math.exp(-b) / y;
	} else {
		tmp = x_m / (y * a);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	tmp = 0
	if (b <= -5.2e+46) or not (b <= 660.0):
		tmp = math.exp(-b) / y
	else:
		tmp = x_m / (y * a)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -5.2e+46) || !(b <= 660.0))
		tmp = Float64(exp(Float64(-b)) / y);
	else
		tmp = Float64(x_m / Float64(y * a));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0;
	if ((b <= -5.2e+46) || ~((b <= 660.0)))
		tmp = exp(-b) / y;
	else
		tmp = x_m / (y * a);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * If[Or[LessEqual[b, -5.2e+46], N[Not[LessEqual[b, 660.0]], $MachinePrecision]], N[(N[Exp[(-b)], $MachinePrecision] / y), $MachinePrecision], N[(x$95$m / N[(y * a), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < -5.20000000000000027e46 or 660 < b

   1. Initial program 100.0%

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

      1. add-exp-log 78.1%

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

      2. *-commutative 78.1%

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

      3. log-prod 51.4%

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

      4. add-log-exp 51.4%

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

      5. associate--l+ 51.4%

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

      6. fma-define 51.4%

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

      7. sub-neg 51.4%

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

      8. metadata-eval 51.4%

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

   4. Applied egg-rr 51.4%

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

   5. Taylor expanded in b around inf 70.7%

      \[\leadsto \frac{e^{\color{blue}{-1 \cdot b}}}{y} \]
   6. Step-by-step derivation

      1. neg-mul-1 70.7%

         \[\leadsto \frac{e^{\color{blue}{-b}}}{y} \]

   7. Simplified 70.7%

      \[\leadsto \frac{e^{\color{blue}{-b}}}{y} \]

   if -5.20000000000000027e46 < b < 660

   1. Initial program 97.1%

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

      1. associate-/l* 96.5%

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

      2. associate--l+ 96.5%

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

      3. exp-sum 84.0%

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

      4. associate-/l* 82.7%

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

      5. *-commutative 82.7%

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

      6. exp-to-pow 82.7%

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

      7. exp-diff 80.7%

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

      8. *-commutative 80.7%

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

      9. exp-to-pow 82.1%

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

      10. sub-neg 82.1%

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

      11. metadata-eval 82.1%

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

   3. Simplified 82.1%

      \[\leadsto \color{blue}{x \cdot \left({z}^{y} \cdot \frac{\frac{{a}^{\left(t + -1\right)}}{e^{b}}}{y}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 66.2%

      \[\leadsto x \cdot \color{blue}{\frac{{z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]

   6. Taylor expanded in b around 0 69.2%

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

      1. associate-/l* 68.9%

         \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{a \cdot y}} \]

   8. Simplified 68.9%

      \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{a \cdot y}} \]

   9. Taylor expanded in y around 0 45.9%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{+46} \lor \neg \left(b \leq 660\right):\\ \;\;\;\;\frac{e^{-b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{x\_m}{a \cdot \left(y \cdot e^{b}\right)} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (* x_s (/ x_m (* a (* y (exp b))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	return x_s * (x_m / (a * (y * exp(b))));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x_s * (x_m / (a * (y * exp(b))))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	return x_s * (x_m / (a * (y * Math.exp(b))));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	return x_s * (x_m / (a * (y * math.exp(b))))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	return Float64(x_s * Float64(x_m / Float64(a * Float64(y * exp(b)))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z, t, a, b)
	tmp = x_s * (x_m / (a * (y * exp(b))));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * N[(x$95$m / N[(a * N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.3%

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

   1. associate-/l* 98.0%

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

   2. associate--l+ 98.0%

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

   3. exp-sum 83.2%

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

   4. associate-/l* 82.4%

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

   5. *-commutative 82.4%

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

   6. exp-to-pow 82.4%

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

   7. exp-diff 73.0%

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

   8. *-commutative 73.0%

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

   9. exp-to-pow 73.8%

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

   10. sub-neg 73.8%

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

   11. metadata-eval 73.8%

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

3. Simplified 73.8%

   \[\leadsto \color{blue}{x \cdot \left({z}^{y} \cdot \frac{\frac{{a}^{\left(t + -1\right)}}{e^{b}}}{y}\right)} \]
4. Add Preprocessing

5. Taylor expanded in t around 0 70.7%

   \[\leadsto x \cdot \color{blue}{\frac{{z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]

6. Taylor expanded in y around 0 62.3%

   \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

7. Final simplification 62.3%

   \[\leadsto \frac{x}{a \cdot \left(y \cdot e^{b}\right)} \]
8. Add Preprocessing

Alternative 14: 40.0% accurate, 11.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{+66}:\\ \;\;\;\;\frac{1}{y} + b \cdot \left(b \cdot \left(-0.16666666666666666 \cdot \frac{b}{y} + 0.5 \cdot \frac{1}{y}\right) + \frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{a}}{y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (*
  x_s
  (if (<= b -2.6e+66)
    (+
     (/ 1.0 y)
     (*
      b
      (+
       (* b (+ (* -0.16666666666666666 (/ b y)) (* 0.5 (/ 1.0 y))))
       (/ -1.0 y))))
    (/ (/ x_m a) y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -2.6e+66) {
		tmp = (1.0 / y) + (b * ((b * ((-0.16666666666666666 * (b / y)) + (0.5 * (1.0 / y)))) + (-1.0 / y)));
	} else {
		tmp = (x_m / a) / y;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-2.6d+66)) then
        tmp = (1.0d0 / y) + (b * ((b * (((-0.16666666666666666d0) * (b / y)) + (0.5d0 * (1.0d0 / y)))) + ((-1.0d0) / y)))
    else
        tmp = (x_m / a) / y
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -2.6e+66) {
		tmp = (1.0 / y) + (b * ((b * ((-0.16666666666666666 * (b / y)) + (0.5 * (1.0 / y)))) + (-1.0 / y)));
	} else {
		tmp = (x_m / a) / y;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	tmp = 0
	if b <= -2.6e+66:
		tmp = (1.0 / y) + (b * ((b * ((-0.16666666666666666 * (b / y)) + (0.5 * (1.0 / y)))) + (-1.0 / y)))
	else:
		tmp = (x_m / a) / y
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0
	if (b <= -2.6e+66)
		tmp = Float64(Float64(1.0 / y) + Float64(b * Float64(Float64(b * Float64(Float64(-0.16666666666666666 * Float64(b / y)) + Float64(0.5 * Float64(1.0 / y)))) + Float64(-1.0 / y))));
	else
		tmp = Float64(Float64(x_m / a) / y);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -2.6e+66)
		tmp = (1.0 / y) + (b * ((b * ((-0.16666666666666666 * (b / y)) + (0.5 * (1.0 / y)))) + (-1.0 / y)));
	else
		tmp = (x_m / a) / y;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * If[LessEqual[b, -2.6e+66], N[(N[(1.0 / y), $MachinePrecision] + N[(b * N[(N[(b * N[(N[(-0.16666666666666666 * N[(b / y), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / a), $MachinePrecision] / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < -2.60000000000000012e66

   1. Initial program 100.0%

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

      1. add-exp-log 56.4%

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

      2. *-commutative 56.4%

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

      3. log-prod 53.8%

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

      4. add-log-exp 53.8%

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

      5. associate--l+ 53.8%

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

      6. fma-define 53.8%

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

      7. sub-neg 53.8%

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

      8. metadata-eval 53.8%

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

   4. Applied egg-rr 53.8%

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

   5. Taylor expanded in b around inf 51.4%

      \[\leadsto \frac{e^{\color{blue}{-1 \cdot b}}}{y} \]
   6. Step-by-step derivation

      1. neg-mul-1 51.4%

         \[\leadsto \frac{e^{\color{blue}{-b}}}{y} \]

   7. Simplified 51.4%

      \[\leadsto \frac{e^{\color{blue}{-b}}}{y} \]

   8. Taylor expanded in b around 0 51.4%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(-0.16666666666666666 \cdot \frac{b}{y} + 0.5 \cdot \frac{1}{y}\right) - \frac{1}{y}\right) + \frac{1}{y}} \]

   if -2.60000000000000012e66 < b

   1. Initial program 98.0%

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

      1. associate-/l* 97.6%

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

      2. associate--l+ 97.6%

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

      3. exp-sum 83.4%

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

      4. associate-/l* 82.4%

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

      5. *-commutative 82.4%

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

      6. exp-to-pow 82.4%

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

      7. exp-diff 74.1%

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

      8. *-commutative 74.1%

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

      9. exp-to-pow 75.1%

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

      10. sub-neg 75.1%

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

      11. metadata-eval 75.1%

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

   3. Simplified 75.1%

      \[\leadsto \color{blue}{x \cdot \left({z}^{y} \cdot \frac{\frac{{a}^{\left(t + -1\right)}}{e^{b}}}{y}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 69.1%

      \[\leadsto x \cdot \color{blue}{\frac{{z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]

   6. Taylor expanded in b around 0 57.4%

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

      1. associate-/l* 57.2%

         \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{a \cdot y}} \]

   8. Simplified 57.2%

      \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{a \cdot y}} \]

   9. Taylor expanded in y around 0 36.6%

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

       1. associate-/r* 37.0%

          \[\leadsto \color{blue}{\frac{\frac{x}{a}}{y}} \]

   11. Simplified 37.0%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{+66}:\\ \;\;\;\;\frac{1}{y} + b \cdot \left(b \cdot \left(-0.16666666666666666 \cdot \frac{b}{y} + 0.5 \cdot \frac{1}{y}\right) + \frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 37.1% accurate, 19.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{+71}:\\ \;\;\;\;\frac{1}{y} + b \cdot \left(b \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{a}}{y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (*
  x_s
  (if (<= b -2.7e+71) (+ (/ 1.0 y) (* b (* b (/ 0.5 y)))) (/ (/ x_m a) y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -2.7e+71) {
		tmp = (1.0 / y) + (b * (b * (0.5 / y)));
	} else {
		tmp = (x_m / a) / y;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-2.7d+71)) then
        tmp = (1.0d0 / y) + (b * (b * (0.5d0 / y)))
    else
        tmp = (x_m / a) / y
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -2.7e+71) {
		tmp = (1.0 / y) + (b * (b * (0.5 / y)));
	} else {
		tmp = (x_m / a) / y;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	tmp = 0
	if b <= -2.7e+71:
		tmp = (1.0 / y) + (b * (b * (0.5 / y)))
	else:
		tmp = (x_m / a) / y
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0
	if (b <= -2.7e+71)
		tmp = Float64(Float64(1.0 / y) + Float64(b * Float64(b * Float64(0.5 / y))));
	else
		tmp = Float64(Float64(x_m / a) / y);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -2.7e+71)
		tmp = (1.0 / y) + (b * (b * (0.5 / y)));
	else
		tmp = (x_m / a) / y;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * If[LessEqual[b, -2.7e+71], N[(N[(1.0 / y), $MachinePrecision] + N[(b * N[(b * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / a), $MachinePrecision] / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < -2.69999999999999997e71

   1. Initial program 100.0%

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

      1. add-exp-log 56.4%

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

      2. *-commutative 56.4%

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

      3. log-prod 53.8%

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

      4. add-log-exp 53.8%

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

      5. associate--l+ 53.8%

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

      6. fma-define 53.8%

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

      7. sub-neg 53.8%

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

      8. metadata-eval 53.8%

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

   4. Applied egg-rr 53.8%

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

   5. Taylor expanded in b around inf 51.4%

      \[\leadsto \frac{e^{\color{blue}{-1 \cdot b}}}{y} \]
   6. Step-by-step derivation

      1. neg-mul-1 51.4%

         \[\leadsto \frac{e^{\color{blue}{-b}}}{y} \]

   7. Simplified 51.4%

      \[\leadsto \frac{e^{\color{blue}{-b}}}{y} \]

   8. Taylor expanded in b around 0 44.2%

      \[\leadsto \color{blue}{b \cdot \left(0.5 \cdot \frac{b}{y} - \frac{1}{y}\right) + \frac{1}{y}} \]

   9. Taylor expanded in b around inf 44.2%

      \[\leadsto b \cdot \color{blue}{\left(0.5 \cdot \frac{b}{y}\right)} + \frac{1}{y} \]
   10. Step-by-step derivation

       1. associate-*r/ 44.2%

          \[\leadsto b \cdot \color{blue}{\frac{0.5 \cdot b}{y}} + \frac{1}{y} \]

       2. *-commutative 44.2%

          \[\leadsto b \cdot \frac{\color{blue}{b \cdot 0.5}}{y} + \frac{1}{y} \]

       3. associate-/l* 44.2%

          \[\leadsto b \cdot \color{blue}{\left(b \cdot \frac{0.5}{y}\right)} + \frac{1}{y} \]

   11. Simplified 44.2%

       \[\leadsto b \cdot \color{blue}{\left(b \cdot \frac{0.5}{y}\right)} + \frac{1}{y} \]

   if -2.69999999999999997e71 < b

   1. Initial program 98.0%

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

      1. associate-/l* 97.6%

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

      2. associate--l+ 97.6%

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

      3. exp-sum 83.4%

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

      4. associate-/l* 82.4%

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

      5. *-commutative 82.4%

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

      6. exp-to-pow 82.4%

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

      7. exp-diff 74.1%

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

      8. *-commutative 74.1%

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

      9. exp-to-pow 75.1%

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

      10. sub-neg 75.1%

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

      11. metadata-eval 75.1%

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

   3. Simplified 75.1%

      \[\leadsto \color{blue}{x \cdot \left({z}^{y} \cdot \frac{\frac{{a}^{\left(t + -1\right)}}{e^{b}}}{y}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 69.1%

      \[\leadsto x \cdot \color{blue}{\frac{{z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]

   6. Taylor expanded in b around 0 57.4%

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

      1. associate-/l* 57.2%

         \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{a \cdot y}} \]

   8. Simplified 57.2%

      \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{a \cdot y}} \]

   9. Taylor expanded in y around 0 36.6%

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

       1. associate-/r* 37.0%

          \[\leadsto \color{blue}{\frac{\frac{x}{a}}{y}} \]

   11. Simplified 37.0%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{+71}:\\ \;\;\;\;\frac{1}{y} + b \cdot \left(b \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 39.2% accurate, 19.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{+69}:\\ \;\;\;\;\frac{1 + b \cdot \left(-1 + b \cdot 0.5\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{a}}{y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (*
  x_s
  (if (<= b -1.15e+69)
    (/ (+ 1.0 (* b (+ -1.0 (* b 0.5)))) y)
    (/ (/ x_m a) y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.15e+69) {
		tmp = (1.0 + (b * (-1.0 + (b * 0.5)))) / y;
	} else {
		tmp = (x_m / a) / y;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.15d+69)) then
        tmp = (1.0d0 + (b * ((-1.0d0) + (b * 0.5d0)))) / y
    else
        tmp = (x_m / a) / y
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.15e+69) {
		tmp = (1.0 + (b * (-1.0 + (b * 0.5)))) / y;
	} else {
		tmp = (x_m / a) / y;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	tmp = 0
	if b <= -1.15e+69:
		tmp = (1.0 + (b * (-1.0 + (b * 0.5)))) / y
	else:
		tmp = (x_m / a) / y
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0
	if (b <= -1.15e+69)
		tmp = Float64(Float64(1.0 + Float64(b * Float64(-1.0 + Float64(b * 0.5)))) / y);
	else
		tmp = Float64(Float64(x_m / a) / y);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -1.15e+69)
		tmp = (1.0 + (b * (-1.0 + (b * 0.5)))) / y;
	else
		tmp = (x_m / a) / y;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * If[LessEqual[b, -1.15e+69], N[(N[(1.0 + N[(b * N[(-1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x$95$m / a), $MachinePrecision] / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < -1.15000000000000008e69

   1. Initial program 100.0%

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

      1. add-exp-log 56.4%

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

      2. *-commutative 56.4%

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

      3. log-prod 53.8%

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

      4. add-log-exp 53.8%

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

      5. associate--l+ 53.8%

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

      6. fma-define 53.8%

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

      7. sub-neg 53.8%

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

      8. metadata-eval 53.8%

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

   4. Applied egg-rr 53.8%

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

   5. Taylor expanded in b around inf 51.4%

      \[\leadsto \frac{e^{\color{blue}{-1 \cdot b}}}{y} \]
   6. Step-by-step derivation

      1. neg-mul-1 51.4%

         \[\leadsto \frac{e^{\color{blue}{-b}}}{y} \]

   7. Simplified 51.4%

      \[\leadsto \frac{e^{\color{blue}{-b}}}{y} \]

   8. Taylor expanded in b around 0 44.2%

      \[\leadsto \color{blue}{b \cdot \left(0.5 \cdot \frac{b}{y} - \frac{1}{y}\right) + \frac{1}{y}} \]

   9. Taylor expanded in y around 0 49.0%

      \[\leadsto \color{blue}{\frac{1 + b \cdot \left(0.5 \cdot b - 1\right)}{y}} \]

   if -1.15000000000000008e69 < b

   1. Initial program 98.0%

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

      1. associate-/l* 97.6%

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

      2. associate--l+ 97.6%

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

      3. exp-sum 83.4%

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

      4. associate-/l* 82.4%

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

      5. *-commutative 82.4%

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

      6. exp-to-pow 82.4%

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

      7. exp-diff 74.1%

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

      8. *-commutative 74.1%

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

      9. exp-to-pow 75.1%

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

      10. sub-neg 75.1%

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

      11. metadata-eval 75.1%

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

   3. Simplified 75.1%

      \[\leadsto \color{blue}{x \cdot \left({z}^{y} \cdot \frac{\frac{{a}^{\left(t + -1\right)}}{e^{b}}}{y}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 69.1%

      \[\leadsto x \cdot \color{blue}{\frac{{z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]

   6. Taylor expanded in b around 0 57.4%

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

      1. associate-/l* 57.2%

         \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{a \cdot y}} \]

   8. Simplified 57.2%

      \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{a \cdot y}} \]

   9. Taylor expanded in y around 0 36.6%

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

       1. associate-/r* 37.0%

          \[\leadsto \color{blue}{\frac{\frac{x}{a}}{y}} \]

   11. Simplified 37.0%

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

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{+69}:\\ \;\;\;\;\frac{1 + b \cdot \left(-1 + b \cdot 0.5\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 33.7% accurate, 22.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 1.02 \cdot 10^{-207}:\\ \;\;\;\;\frac{x\_m}{a} \cdot \frac{1 - b}{y}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{\frac{1}{a}}{y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (*
  x_s
  (if (<= b 1.02e-207) (* (/ x_m a) (/ (- 1.0 b) y)) (* x_m (/ (/ 1.0 a) y)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= 1.02e-207) {
		tmp = (x_m / a) * ((1.0 - b) / y);
	} else {
		tmp = x_m * ((1.0 / a) / y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.02d-207) then
        tmp = (x_m / a) * ((1.0d0 - b) / y)
    else
        tmp = x_m * ((1.0d0 / a) / y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= 1.02e-207) {
		tmp = (x_m / a) * ((1.0 - b) / y);
	} else {
		tmp = x_m * ((1.0 / a) / y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	tmp = 0
	if b <= 1.02e-207:
		tmp = (x_m / a) * ((1.0 - b) / y)
	else:
		tmp = x_m * ((1.0 / a) / y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0
	if (b <= 1.02e-207)
		tmp = Float64(Float64(x_m / a) * Float64(Float64(1.0 - b) / y));
	else
		tmp = Float64(x_m * Float64(Float64(1.0 / a) / y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0;
	if (b <= 1.02e-207)
		tmp = (x_m / a) * ((1.0 - b) / y);
	else
		tmp = x_m * ((1.0 / a) / y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * If[LessEqual[b, 1.02e-207], N[(N[(x$95$m / a), $MachinePrecision] * N[(N[(1.0 - b), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(1.0 / a), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 1.02e-207

   1. Initial program 97.2%

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

      1. add-exp-log 65.0%

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

      2. *-commutative 65.0%

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

      3. log-prod 46.6%

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

      4. add-log-exp 46.6%

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

      5. associate--l+ 46.6%

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

      6. fma-define 46.6%

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

      7. sub-neg 46.6%

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

      8. metadata-eval 46.6%

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

   4. Applied egg-rr 46.6%

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

   5. Taylor expanded in y around 0 43.0%

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

   6. Taylor expanded in b around 0 32.3%

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

      1. associate-*r* 32.3%

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

      2. neg-mul-1 32.3%

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

      3. distribute-rgt1-in 35.7%

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

      4. exp-sum 35.8%

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

      5. rem-exp-log 69.1%

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

      6. exp-to-pow 70.4%

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

      7. sub-neg 70.4%

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

      8. metadata-eval 70.4%

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

   8. Simplified 70.4%

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

   9. Taylor expanded in t around 0 43.4%

      \[\leadsto \color{blue}{\frac{x \cdot \left(1 - b\right)}{a \cdot y}} \]
   10. Step-by-step derivation

       1. times-frac 47.6%

          \[\leadsto \color{blue}{\frac{x}{a} \cdot \frac{1 - b}{y}} \]

   11. Simplified 47.6%

       \[\leadsto \color{blue}{\frac{x}{a} \cdot \frac{1 - b}{y}} \]

   if 1.02e-207 < b

   1. Initial program 99.7%

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

      1. associate-/l* 99.7%

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

      2. associate--l+ 99.7%

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

      3. exp-sum 85.1%

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

      4. associate-/l* 85.1%

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

      5. *-commutative 85.1%

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

      6. exp-to-pow 85.1%

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

      7. exp-diff 71.3%

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

      8. *-commutative 71.3%

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

      9. exp-to-pow 71.5%

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

      10. sub-neg 71.5%

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

      11. metadata-eval 71.5%

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

   3. Simplified 71.5%

      \[\leadsto \color{blue}{x \cdot \left({z}^{y} \cdot \frac{\frac{{a}^{\left(t + -1\right)}}{e^{b}}}{y}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 75.5%

      \[\leadsto x \cdot \color{blue}{\frac{{z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]

   6. Taylor expanded in b around 0 48.9%

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

      1. associate-/l* 48.9%

         \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{a \cdot y}} \]

   8. Simplified 48.9%

      \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{a \cdot y}} \]

   9. Taylor expanded in y around 0 28.1%

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

       1. associate-/r* 28.1%

          \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{a}}{y}} \]

   11. Simplified 28.1%

       \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{a}}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.02 \cdot 10^{-207}:\\ \;\;\;\;\frac{x}{a} \cdot \frac{1 - b}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{1}{a}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 32.0% accurate, 31.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+90}:\\ \;\;\;\;\frac{1 - b}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y \cdot a}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (* x_s (if (<= b -1.6e+90) (/ (- 1.0 b) y) (/ x_m (* y a)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.6e+90) {
		tmp = (1.0 - b) / y;
	} else {
		tmp = x_m / (y * a);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.6d+90)) then
        tmp = (1.0d0 - b) / y
    else
        tmp = x_m / (y * a)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.6e+90) {
		tmp = (1.0 - b) / y;
	} else {
		tmp = x_m / (y * a);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	tmp = 0
	if b <= -1.6e+90:
		tmp = (1.0 - b) / y
	else:
		tmp = x_m / (y * a)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0
	if (b <= -1.6e+90)
		tmp = Float64(Float64(1.0 - b) / y);
	else
		tmp = Float64(x_m / Float64(y * a));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -1.6e+90)
		tmp = (1.0 - b) / y;
	else
		tmp = x_m / (y * a);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * If[LessEqual[b, -1.6e+90], N[(N[(1.0 - b), $MachinePrecision] / y), $MachinePrecision], N[(x$95$m / N[(y * a), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < -1.59999999999999999e90

   1. Initial program 100.0%

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

      1. add-exp-log 61.1%

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

      2. *-commutative 61.1%

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

      3. log-prod 58.3%

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

      4. add-log-exp 58.3%

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

      5. associate--l+ 58.3%

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

      6. fma-define 58.3%

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

      7. sub-neg 58.3%

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

      8. metadata-eval 58.3%

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

   4. Applied egg-rr 58.3%

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

   5. Taylor expanded in b around inf 55.6%

      \[\leadsto \frac{e^{\color{blue}{-1 \cdot b}}}{y} \]
   6. Step-by-step derivation

      1. neg-mul-1 55.6%

         \[\leadsto \frac{e^{\color{blue}{-b}}}{y} \]

   7. Simplified 55.6%

      \[\leadsto \frac{e^{\color{blue}{-b}}}{y} \]

   8. Taylor expanded in b around 0 47.8%

      \[\leadsto \color{blue}{b \cdot \left(0.5 \cdot \frac{b}{y} - \frac{1}{y}\right) + \frac{1}{y}} \]

   9. Taylor expanded in b around 0 26.9%

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

       1. +-commutative 26.9%

          \[\leadsto \color{blue}{\frac{1}{y} + -1 \cdot \frac{b}{y}} \]

       2. mul-1-neg 26.9%

          \[\leadsto \frac{1}{y} + \color{blue}{\left(-\frac{b}{y}\right)} \]

       3. unsub-neg 26.9%

          \[\leadsto \color{blue}{\frac{1}{y} - \frac{b}{y}} \]

       4. div-sub 26.9%

          \[\leadsto \color{blue}{\frac{1 - b}{y}} \]

   11. Simplified 26.9%

       \[\leadsto \color{blue}{\frac{1 - b}{y}} \]

   if -1.59999999999999999e90 < b

   1. Initial program 98.0%

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

      1. associate-/l* 97.6%

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

      2. associate--l+ 97.6%

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

      3. exp-sum 83.1%

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

      4. associate-/l* 82.2%

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

      5. *-commutative 82.2%

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

      6. exp-to-pow 82.2%

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

      7. exp-diff 74.0%

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

      8. *-commutative 74.0%

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

      9. exp-to-pow 75.0%

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

      10. sub-neg 75.0%

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

      11. metadata-eval 75.0%

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

   3. Simplified 75.0%

      \[\leadsto \color{blue}{x \cdot \left({z}^{y} \cdot \frac{\frac{{a}^{\left(t + -1\right)}}{e^{b}}}{y}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 69.1%

      \[\leadsto x \cdot \color{blue}{\frac{{z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]

   6. Taylor expanded in b around 0 57.1%

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

      1. associate-/l* 56.9%

         \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{a \cdot y}} \]

   8. Simplified 56.9%

      \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{a \cdot y}} \]

   9. Taylor expanded in y around 0 36.2%

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

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+90}:\\ \;\;\;\;\frac{1 - b}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 31.9% accurate, 31.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+86}:\\ \;\;\;\;\frac{1 - b}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{a}}{y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (* x_s (if (<= b -4.5e+86) (/ (- 1.0 b) y) (/ (/ x_m a) y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -4.5e+86) {
		tmp = (1.0 - b) / y;
	} else {
		tmp = (x_m / a) / y;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-4.5d+86)) then
        tmp = (1.0d0 - b) / y
    else
        tmp = (x_m / a) / y
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -4.5e+86) {
		tmp = (1.0 - b) / y;
	} else {
		tmp = (x_m / a) / y;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	tmp = 0
	if b <= -4.5e+86:
		tmp = (1.0 - b) / y
	else:
		tmp = (x_m / a) / y
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0
	if (b <= -4.5e+86)
		tmp = Float64(Float64(1.0 - b) / y);
	else
		tmp = Float64(Float64(x_m / a) / y);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -4.5e+86)
		tmp = (1.0 - b) / y;
	else
		tmp = (x_m / a) / y;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * If[LessEqual[b, -4.5e+86], N[(N[(1.0 - b), $MachinePrecision] / y), $MachinePrecision], N[(N[(x$95$m / a), $MachinePrecision] / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < -4.49999999999999993e86

   1. Initial program 100.0%

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

      1. add-exp-log 61.1%

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

      2. *-commutative 61.1%

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

      3. log-prod 58.3%

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

      4. add-log-exp 58.3%

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

      5. associate--l+ 58.3%

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

      6. fma-define 58.3%

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

      7. sub-neg 58.3%

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

      8. metadata-eval 58.3%

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

   4. Applied egg-rr 58.3%

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

   5. Taylor expanded in b around inf 55.6%

      \[\leadsto \frac{e^{\color{blue}{-1 \cdot b}}}{y} \]
   6. Step-by-step derivation

      1. neg-mul-1 55.6%

         \[\leadsto \frac{e^{\color{blue}{-b}}}{y} \]

   7. Simplified 55.6%

      \[\leadsto \frac{e^{\color{blue}{-b}}}{y} \]

   8. Taylor expanded in b around 0 47.8%

      \[\leadsto \color{blue}{b \cdot \left(0.5 \cdot \frac{b}{y} - \frac{1}{y}\right) + \frac{1}{y}} \]

   9. Taylor expanded in b around 0 26.9%

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

       1. +-commutative 26.9%

          \[\leadsto \color{blue}{\frac{1}{y} + -1 \cdot \frac{b}{y}} \]

       2. mul-1-neg 26.9%

          \[\leadsto \frac{1}{y} + \color{blue}{\left(-\frac{b}{y}\right)} \]

       3. unsub-neg 26.9%

          \[\leadsto \color{blue}{\frac{1}{y} - \frac{b}{y}} \]

       4. div-sub 26.9%

          \[\leadsto \color{blue}{\frac{1 - b}{y}} \]

   11. Simplified 26.9%

       \[\leadsto \color{blue}{\frac{1 - b}{y}} \]

   if -4.49999999999999993e86 < b

   1. Initial program 98.0%

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

      1. associate-/l* 97.6%

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

      2. associate--l+ 97.6%

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

      3. exp-sum 83.1%

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

      4. associate-/l* 82.2%

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

      5. *-commutative 82.2%

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

      6. exp-to-pow 82.2%

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

      7. exp-diff 74.0%

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

      8. *-commutative 74.0%

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

      9. exp-to-pow 75.0%

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

      10. sub-neg 75.0%

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

      11. metadata-eval 75.0%

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

   3. Simplified 75.0%

      \[\leadsto \color{blue}{x \cdot \left({z}^{y} \cdot \frac{\frac{{a}^{\left(t + -1\right)}}{e^{b}}}{y}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 69.1%

      \[\leadsto x \cdot \color{blue}{\frac{{z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]

   6. Taylor expanded in b around 0 57.1%

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

      1. associate-/l* 56.9%

         \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{a \cdot y}} \]

   8. Simplified 56.9%

      \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{a \cdot y}} \]

   9. Taylor expanded in y around 0 36.2%

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

       1. associate-/r* 36.5%

          \[\leadsto \color{blue}{\frac{\frac{x}{a}}{y}} \]

   11. Simplified 36.5%

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

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+86}:\\ \;\;\;\;\frac{1 - b}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 31.0% accurate, 63.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{x\_m}{y \cdot a} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b) :precision binary64 (* x_s (/ x_m (* y a))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	return x_s * (x_m / (y * a));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x_s * (x_m / (y * a))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	return x_s * (x_m / (y * a));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	return x_s * (x_m / (y * a))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	return Float64(x_s * Float64(x_m / Float64(y * a)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z, t, a, b)
	tmp = x_s * (x_m / (y * a));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * N[(x$95$m / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.3%

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

   1. associate-/l* 98.0%

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

   2. associate--l+ 98.0%

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

   3. exp-sum 83.2%

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

   4. associate-/l* 82.4%

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

   5. *-commutative 82.4%

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

   6. exp-to-pow 82.4%

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

   7. exp-diff 73.0%

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

   8. *-commutative 73.0%

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

   9. exp-to-pow 73.8%

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

   10. sub-neg 73.8%

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

   11. metadata-eval 73.8%

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

3. Simplified 73.8%

   \[\leadsto \color{blue}{x \cdot \left({z}^{y} \cdot \frac{\frac{{a}^{\left(t + -1\right)}}{e^{b}}}{y}\right)} \]
4. Add Preprocessing

5. Taylor expanded in t around 0 70.7%

   \[\leadsto x \cdot \color{blue}{\frac{{z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]

6. Taylor expanded in b around 0 52.6%

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

   1. associate-/l* 52.7%

      \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{a \cdot y}} \]

8. Simplified 52.7%

   \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{a \cdot y}} \]

9. Taylor expanded in y around 0 33.4%

   \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]

10. Final simplification 33.4%

    \[\leadsto \frac{x}{y \cdot a} \]
11. Add Preprocessing

Alternative 21: 4.0% accurate, 105.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{1}{y} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b) :precision binary64 (* x_s (/ 1.0 y)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	return x_s * (1.0 / y);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x_s * (1.0d0 / y)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	return x_s * (1.0 / y);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	return x_s * (1.0 / y)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	return Float64(x_s * Float64(1.0 / y))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z, t, a, b)
	tmp = x_s * (1.0 / y);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.3%

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

   1. add-exp-log 74.1%

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

   2. *-commutative 74.1%

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

   3. log-prod 50.2%

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

   4. add-log-exp 50.2%

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

   5. associate--l+ 50.2%

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

   6. fma-define 50.2%

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

   7. sub-neg 50.2%

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

   8. metadata-eval 50.2%

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

4. Applied egg-rr 50.2%

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

5. Taylor expanded in b around inf 32.0%

   \[\leadsto \frac{e^{\color{blue}{-1 \cdot b}}}{y} \]
6. Step-by-step derivation

   1. neg-mul-1 32.0%

      \[\leadsto \frac{e^{\color{blue}{-b}}}{y} \]

7. Simplified 32.0%

   \[\leadsto \frac{e^{\color{blue}{-b}}}{y} \]

8. Taylor expanded in b around 0 3.2%

   \[\leadsto \color{blue}{\frac{1}{y}} \]

9. Final simplification 3.2%

   \[\leadsto \frac{1}{y} \]
10. Add Preprocessing

Developer target: 72.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {a}^{\left(t - 1\right)}\\ t_2 := \frac{x \cdot \frac{t\_1}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{if}\;t < -0.8845848504127471:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < 852031.2288374073:\\ \;\;\;\;\frac{\frac{x}{y} \cdot t\_1}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (pow a (- t 1.0)))
        (t_2 (/ (* x (/ t_1 y)) (- (+ b 1.0) (* y (log z))))))
   (if (< t -0.8845848504127471)
     t_2
     (if (< t 852031.2288374073)
       (/ (* (/ x y) t_1) (exp (- b (* (log z) y))))
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = pow(a, (t - 1.0));
	double t_2 = (x * (t_1 / y)) / ((b + 1.0) - (y * log(z)));
	double tmp;
	if (t < -0.8845848504127471) {
		tmp = t_2;
	} else if (t < 852031.2288374073) {
		tmp = ((x / y) * t_1) / exp((b - (log(z) * y)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = a ** (t - 1.0d0)
    t_2 = (x * (t_1 / y)) / ((b + 1.0d0) - (y * log(z)))
    if (t < (-0.8845848504127471d0)) then
        tmp = t_2
    else if (t < 852031.2288374073d0) then
        tmp = ((x / y) * t_1) / exp((b - (log(z) * y)))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.pow(a, (t - 1.0));
	double t_2 = (x * (t_1 / y)) / ((b + 1.0) - (y * Math.log(z)));
	double tmp;
	if (t < -0.8845848504127471) {
		tmp = t_2;
	} else if (t < 852031.2288374073) {
		tmp = ((x / y) * t_1) / Math.exp((b - (Math.log(z) * y)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.pow(a, (t - 1.0))
	t_2 = (x * (t_1 / y)) / ((b + 1.0) - (y * math.log(z)))
	tmp = 0
	if t < -0.8845848504127471:
		tmp = t_2
	elif t < 852031.2288374073:
		tmp = ((x / y) * t_1) / math.exp((b - (math.log(z) * y)))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = a ^ Float64(t - 1.0)
	t_2 = Float64(Float64(x * Float64(t_1 / y)) / Float64(Float64(b + 1.0) - Float64(y * log(z))))
	tmp = 0.0
	if (t < -0.8845848504127471)
		tmp = t_2;
	elseif (t < 852031.2288374073)
		tmp = Float64(Float64(Float64(x / y) * t_1) / exp(Float64(b - Float64(log(z) * y))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a ^ (t - 1.0);
	t_2 = (x * (t_1 / y)) / ((b + 1.0) - (y * log(z)));
	tmp = 0.0;
	if (t < -0.8845848504127471)
		tmp = t_2;
	elseif (t < 852031.2288374073)
		tmp = ((x / y) * t_1) / exp((b - (log(z) * y)));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision] / N[(N[(b + 1.0), $MachinePrecision] - N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -0.8845848504127471], t$95$2, If[Less[t, 852031.2288374073], N[(N[(N[(x / y), $MachinePrecision] * t$95$1), $MachinePrecision] / N[Exp[N[(b - N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2024131 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (if (< t -0.8845848504127471) (/ (* x (/ (pow a (- t 1.0)) y)) (- (+ b 1.0) (* y (log z)))) (if (< t 852031.2288374073) (/ (* (/ x y) (pow a (- t 1.0))) (exp (- b (* (log z) y)))) (/ (* x (/ (pow a (- t 1.0)) y)) (- (+ b 1.0) (* y (log z))))))

  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))