Harley's example

Percentage Accurate: 91.3% → 96.7%

Time: 38.4s

Alternatives: 5

Speedup: 8.7×

• could not determine a ground truth

  1. c_p = 2.6207644310957015e-248
  2. c_n = 1.0784803993510114e+63
  3. t = -1.1511978279313845e-57
  4. s = -4.093742473117237e-205

Specification

\[0 < c\_p \land 0 < c\_n\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{1 + e^{-t}}\\ t_2 := \frac{1}{1 + e^{-s}}\\ \frac{{t\_2}^{c\_p} \cdot {\left(1 - t\_2\right)}^{c\_n}}{{t\_1}^{c\_p} \cdot {\left(1 - t\_1\right)}^{c\_n}} \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (/ 1.0 (+ 1.0 (exp (- t))))) (t_2 (/ 1.0 (+ 1.0 (exp (- s))))))
   (/
    (* (pow t_2 c_p) (pow (- 1.0 t_2) c_n))
    (* (pow t_1 c_p) (pow (- 1.0 t_1) c_n)))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = 1.0 / (1.0 + exp(-t));
	double t_2 = 1.0 / (1.0 + exp(-s));
	return (pow(t_2, c_p) * pow((1.0 - t_2), c_n)) / (pow(t_1, c_p) * pow((1.0 - t_1), c_n));
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    real(8) :: t_1
    real(8) :: t_2
    t_1 = 1.0d0 / (1.0d0 + exp(-t))
    t_2 = 1.0d0 / (1.0d0 + exp(-s))
    code = ((t_2 ** c_p) * ((1.0d0 - t_2) ** c_n)) / ((t_1 ** c_p) * ((1.0d0 - t_1) ** c_n))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double t_1 = 1.0 / (1.0 + Math.exp(-t));
	double t_2 = 1.0 / (1.0 + Math.exp(-s));
	return (Math.pow(t_2, c_p) * Math.pow((1.0 - t_2), c_n)) / (Math.pow(t_1, c_p) * Math.pow((1.0 - t_1), c_n));
}

Python

def code(c_p, c_n, t, s):
	t_1 = 1.0 / (1.0 + math.exp(-t))
	t_2 = 1.0 / (1.0 + math.exp(-s))
	return (math.pow(t_2, c_p) * math.pow((1.0 - t_2), c_n)) / (math.pow(t_1, c_p) * math.pow((1.0 - t_1), c_n))

Julia

function code(c_p, c_n, t, s)
	t_1 = Float64(1.0 / Float64(1.0 + exp(Float64(-t))))
	t_2 = Float64(1.0 / Float64(1.0 + exp(Float64(-s))))
	return Float64(Float64((t_2 ^ c_p) * (Float64(1.0 - t_2) ^ c_n)) / Float64((t_1 ^ c_p) * (Float64(1.0 - t_1) ^ c_n)))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	t_1 = 1.0 / (1.0 + exp(-t));
	t_2 = 1.0 / (1.0 + exp(-s));
	tmp = ((t_2 ^ c_p) * ((1.0 - t_2) ^ c_n)) / ((t_1 ^ c_p) * ((1.0 - t_1) ^ c_n));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[(1.0 / N[(1.0 + N[Exp[(-t)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(1.0 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[t$95$2, c$95$p], $MachinePrecision] * N[Power[N[(1.0 - t$95$2), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$1, c$95$p], $MachinePrecision] * N[Power[N[(1.0 - t$95$1), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Initial Program: 91.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{1 + e^{-t}}\\ t_2 := \frac{1}{1 + e^{-s}}\\ \frac{{t\_2}^{c\_p} \cdot {\left(1 - t\_2\right)}^{c\_n}}{{t\_1}^{c\_p} \cdot {\left(1 - t\_1\right)}^{c\_n}} \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (/ 1.0 (+ 1.0 (exp (- t))))) (t_2 (/ 1.0 (+ 1.0 (exp (- s))))))
   (/
    (* (pow t_2 c_p) (pow (- 1.0 t_2) c_n))
    (* (pow t_1 c_p) (pow (- 1.0 t_1) c_n)))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = 1.0 / (1.0 + exp(-t));
	double t_2 = 1.0 / (1.0 + exp(-s));
	return (pow(t_2, c_p) * pow((1.0 - t_2), c_n)) / (pow(t_1, c_p) * pow((1.0 - t_1), c_n));
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    real(8) :: t_1
    real(8) :: t_2
    t_1 = 1.0d0 / (1.0d0 + exp(-t))
    t_2 = 1.0d0 / (1.0d0 + exp(-s))
    code = ((t_2 ** c_p) * ((1.0d0 - t_2) ** c_n)) / ((t_1 ** c_p) * ((1.0d0 - t_1) ** c_n))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double t_1 = 1.0 / (1.0 + Math.exp(-t));
	double t_2 = 1.0 / (1.0 + Math.exp(-s));
	return (Math.pow(t_2, c_p) * Math.pow((1.0 - t_2), c_n)) / (Math.pow(t_1, c_p) * Math.pow((1.0 - t_1), c_n));
}

Python

def code(c_p, c_n, t, s):
	t_1 = 1.0 / (1.0 + math.exp(-t))
	t_2 = 1.0 / (1.0 + math.exp(-s))
	return (math.pow(t_2, c_p) * math.pow((1.0 - t_2), c_n)) / (math.pow(t_1, c_p) * math.pow((1.0 - t_1), c_n))

Julia

function code(c_p, c_n, t, s)
	t_1 = Float64(1.0 / Float64(1.0 + exp(Float64(-t))))
	t_2 = Float64(1.0 / Float64(1.0 + exp(Float64(-s))))
	return Float64(Float64((t_2 ^ c_p) * (Float64(1.0 - t_2) ^ c_n)) / Float64((t_1 ^ c_p) * (Float64(1.0 - t_1) ^ c_n)))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	t_1 = 1.0 / (1.0 + exp(-t));
	t_2 = 1.0 / (1.0 + exp(-s));
	tmp = ((t_2 ^ c_p) * ((1.0 - t_2) ^ c_n)) / ((t_1 ^ c_p) * ((1.0 - t_1) ^ c_n));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[(1.0 / N[(1.0 + N[Exp[(-t)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(1.0 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[t$95$2, c$95$p], $MachinePrecision] * N[Power[N[(1.0 - t$95$2), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$1, c$95$p], $MachinePrecision] * N[Power[N[(1.0 - t$95$1), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{1 + e^{-t}}\\ t_2 := e^{-s}\\ t_3 := \frac{1}{1 + t\_2}\\ t_4 := \frac{{t\_3}^{c\_p} \cdot {\left(1 - t\_3\right)}^{c\_n}}{{t\_1}^{c\_p} \cdot {\left(1 - t\_1\right)}^{c\_n}}\\ \mathbf{if}\;t\_4 \leq 2:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (/ 1.0 (+ 1.0 (exp (- t)))))
        (t_2 (exp (- s)))
        (t_3 (/ 1.0 (+ 1.0 t_2)))
        (t_4
         (/
          (* (pow t_3 c_p) (pow (- 1.0 t_3) c_n))
          (* (pow t_1 c_p) (pow (- 1.0 t_1) c_n)))))
   (if (<= t_4 2.0) t_4 t_2)))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = 1.0 / (1.0 + exp(-t));
	double t_2 = exp(-s);
	double t_3 = 1.0 / (1.0 + t_2);
	double t_4 = (pow(t_3, c_p) * pow((1.0 - t_3), c_n)) / (pow(t_1, c_p) * pow((1.0 - t_1), c_n));
	double tmp;
	if (t_4 <= 2.0) {
		tmp = t_4;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = 1.0d0 / (1.0d0 + exp(-t))
    t_2 = exp(-s)
    t_3 = 1.0d0 / (1.0d0 + t_2)
    t_4 = ((t_3 ** c_p) * ((1.0d0 - t_3) ** c_n)) / ((t_1 ** c_p) * ((1.0d0 - t_1) ** c_n))
    if (t_4 <= 2.0d0) then
        tmp = t_4
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double t_1 = 1.0 / (1.0 + Math.exp(-t));
	double t_2 = Math.exp(-s);
	double t_3 = 1.0 / (1.0 + t_2);
	double t_4 = (Math.pow(t_3, c_p) * Math.pow((1.0 - t_3), c_n)) / (Math.pow(t_1, c_p) * Math.pow((1.0 - t_1), c_n));
	double tmp;
	if (t_4 <= 2.0) {
		tmp = t_4;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	t_1 = 1.0 / (1.0 + math.exp(-t))
	t_2 = math.exp(-s)
	t_3 = 1.0 / (1.0 + t_2)
	t_4 = (math.pow(t_3, c_p) * math.pow((1.0 - t_3), c_n)) / (math.pow(t_1, c_p) * math.pow((1.0 - t_1), c_n))
	tmp = 0
	if t_4 <= 2.0:
		tmp = t_4
	else:
		tmp = t_2
	return tmp

Julia

function code(c_p, c_n, t, s)
	t_1 = Float64(1.0 / Float64(1.0 + exp(Float64(-t))))
	t_2 = exp(Float64(-s))
	t_3 = Float64(1.0 / Float64(1.0 + t_2))
	t_4 = Float64(Float64((t_3 ^ c_p) * (Float64(1.0 - t_3) ^ c_n)) / Float64((t_1 ^ c_p) * (Float64(1.0 - t_1) ^ c_n)))
	tmp = 0.0
	if (t_4 <= 2.0)
		tmp = t_4;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	t_1 = 1.0 / (1.0 + exp(-t));
	t_2 = exp(-s);
	t_3 = 1.0 / (1.0 + t_2);
	t_4 = ((t_3 ^ c_p) * ((1.0 - t_3) ^ c_n)) / ((t_1 ^ c_p) * ((1.0 - t_1) ^ c_n));
	tmp = 0.0;
	if (t_4 <= 2.0)
		tmp = t_4;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[(1.0 / N[(1.0 + N[Exp[(-t)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-s)], $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Power[t$95$3, c$95$p], $MachinePrecision] * N[Power[N[(1.0 - t$95$3), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$1, c$95$p], $MachinePrecision] * N[Power[N[(1.0 - t$95$1), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 2.0], t$95$4, t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s))))) c_n)) (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t))))) c_n))) < 2

   1. Initial program 98.4%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. Add Preprocessing

   if 2 < (/.f64 (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s))))) c_n)) (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t))))) c_n)))

   1. Initial program 0.2%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. Add Preprocessing

   3. Taylor expanded in c_p around 0

      \[\leadsto \color{blue}{c\_p \cdot \left(\left(c\_p \cdot \left(\frac{1}{2} \cdot \frac{{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} - \left(\frac{1}{2} \cdot \frac{{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{2} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} + \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot \left(\frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} - \frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}\right)\right)\right) + \frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}\right) - \frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}\right) + \frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

   5. Applied rewrites 56.9%

      \[\leadsto \color{blue}{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}} \]

   6. Taylor expanded in c_p around 0

      \[\leadsto 1 + \color{blue}{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)} \]

   8. Applied rewrites 80.3%

      \[\leadsto e^{-s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 96.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{-s}\\ \mathbf{if}\;-s \leq 2 \cdot 10^{-53}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{1 + t\_1}\right)}^{c\_p}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (exp (- s))))
   (if (<= (- s) 2e-53) t_1 (pow (/ 1.0 (+ 1.0 t_1)) c_p))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = exp(-s);
	double tmp;
	if (-s <= 2e-53) {
		tmp = t_1;
	} else {
		tmp = pow((1.0 / (1.0 + t_1)), c_p);
	}
	return tmp;
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    real(8) :: t_1
    real(8) :: tmp
    t_1 = exp(-s)
    if (-s <= 2d-53) then
        tmp = t_1
    else
        tmp = (1.0d0 / (1.0d0 + t_1)) ** c_p
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double t_1 = Math.exp(-s);
	double tmp;
	if (-s <= 2e-53) {
		tmp = t_1;
	} else {
		tmp = Math.pow((1.0 / (1.0 + t_1)), c_p);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	t_1 = math.exp(-s)
	tmp = 0
	if -s <= 2e-53:
		tmp = t_1
	else:
		tmp = math.pow((1.0 / (1.0 + t_1)), c_p)
	return tmp

Julia

function code(c_p, c_n, t, s)
	t_1 = exp(Float64(-s))
	tmp = 0.0
	if (Float64(-s) <= 2e-53)
		tmp = t_1;
	else
		tmp = Float64(1.0 / Float64(1.0 + t_1)) ^ c_p;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	t_1 = exp(-s);
	tmp = 0.0;
	if (-s <= 2e-53)
		tmp = t_1;
	else
		tmp = (1.0 / (1.0 + t_1)) ^ c_p;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[Exp[(-s)], $MachinePrecision]}, If[LessEqual[(-s), 2e-53], t$95$1, N[Power[N[(1.0 / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 s) < 2.00000000000000006e-53

   1. Initial program 93.7%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. Add Preprocessing

   3. Taylor expanded in c_p around 0

      \[\leadsto \color{blue}{c\_p \cdot \left(\left(c\_p \cdot \left(\frac{1}{2} \cdot \frac{{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} - \left(\frac{1}{2} \cdot \frac{{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{2} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} + \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot \left(\frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} - \frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}\right)\right)\right) + \frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}\right) - \frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}\right) + \frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

   5. Applied rewrites 93.0%

      \[\leadsto \color{blue}{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}} \]

   6. Taylor expanded in c_p around 0

      \[\leadsto 1 + \color{blue}{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)} \]

   8. Applied rewrites 96.7%

      \[\leadsto e^{-s} \]

   if 2.00000000000000006e-53 < (neg.f64 s)

   1. Initial program 80.3%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. Add Preprocessing

   3. Taylor expanded in c_p around 0

      \[\leadsto \color{blue}{c\_p \cdot \left(\left(c\_p \cdot \left(\frac{1}{2} \cdot \frac{{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} - \left(\frac{1}{2} \cdot \frac{{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{2} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} + \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot \left(\frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} - \frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}\right)\right)\right) + \frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}\right) - \frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}\right) + \frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

   5. Applied rewrites 90.3%

      \[\leadsto \color{blue}{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 93.5% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-t \leq 4 \cdot 10^{-37}:\\ \;\;\;\;e^{-t}\\ \mathbf{else}:\\ \;\;\;\;e^{-s}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- t) 4e-37) (exp (- t)) (exp (- s))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-t <= 4e-37) {
		tmp = exp(-t);
	} else {
		tmp = exp(-s);
	}
	return tmp;
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    real(8) :: tmp
    if (-t <= 4d-37) then
        tmp = exp(-t)
    else
        tmp = exp(-s)
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-t <= 4e-37) {
		tmp = Math.exp(-t);
	} else {
		tmp = Math.exp(-s);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if -t <= 4e-37:
		tmp = math.exp(-t)
	else:
		tmp = math.exp(-s)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (Float64(-t) <= 4e-37)
		tmp = exp(Float64(-t));
	else
		tmp = exp(Float64(-s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (-t <= 4e-37)
		tmp = exp(-t);
	else
		tmp = exp(-s);
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-t), 4e-37], N[Exp[(-t)], $MachinePrecision], N[Exp[(-s)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 t) < 4.00000000000000027e-37

   1. Initial program 92.8%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. Add Preprocessing

   3. Taylor expanded in c_p around 0

      \[\leadsto \color{blue}{c\_p \cdot \left(\left(c\_p \cdot \left(\frac{1}{2} \cdot \frac{{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} - \left(\frac{1}{2} \cdot \frac{{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{2} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} + \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot \left(\frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} - \frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}\right)\right)\right) + \frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}\right) - \frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}\right) + \frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

   5. Applied rewrites 93.6%

      \[\leadsto \color{blue}{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}} \]

   6. Taylor expanded in c_p around 0

      \[\leadsto 1 + \color{blue}{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)} \]

   8. Applied rewrites 93.0%

      \[\leadsto e^{-s} \]

   9. Taylor expanded in c_p around 0

      \[\leadsto \color{blue}{c\_p \cdot \left(\left(c\_p \cdot \left(\left(\frac{1}{2} \cdot \frac{{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} + c\_p \cdot \left(\frac{1}{6} \cdot \frac{{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{3} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} - \left(\frac{1}{6} \cdot \frac{{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{3} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} + \left(\frac{1}{2} \cdot \left({\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{2} \cdot \left(\frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} - \frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}\right)\right) + \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot \left(\frac{1}{2} \cdot \frac{{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} - \left(\frac{1}{2} \cdot \frac{{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{2} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} + \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot \left(\frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} - \frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}\right)\right)\right)\right)\right)\right)\right) - \left(\frac{1}{2} \cdot \frac{{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{2} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} + \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot \left(\frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} - \frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}\right)\right)\right) + \frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}\right) - \frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}\right) + \frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

   11. Applied rewrites 3.6%

       \[\leadsto \color{blue}{-s} \]

   12. Taylor expanded in c_n around 0

       \[\leadsto \color{blue}{\frac{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}}} \]

   14. Applied rewrites 95.8%

       \[\leadsto \color{blue}{e^{-t}} \]

   if 4.00000000000000027e-37 < (neg.f64 t)

   1. Initial program 91.4%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. Add Preprocessing

   3. Taylor expanded in c_p around 0

      \[\leadsto \color{blue}{c\_p \cdot \left(\left(c\_p \cdot \left(\frac{1}{2} \cdot \frac{{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} - \left(\frac{1}{2} \cdot \frac{{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{2} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} + \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot \left(\frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} - \frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}\right)\right)\right) + \frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}\right) - \frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}\right) + \frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

   5. Applied rewrites 84.4%

      \[\leadsto \color{blue}{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}} \]

   6. Taylor expanded in c_p around 0

      \[\leadsto 1 + \color{blue}{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)} \]

   8. Applied rewrites 95.1%

      \[\leadsto e^{-s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 92.9% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ e^{-s} \end{array} \]

FPCore

(FPCore (c_p c_n t s) :precision binary64 (exp (- s)))

C

double code(double c_p, double c_n, double t, double s) {
	return exp(-s);
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    code = exp(-s)
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return Math.exp(-s);
}

Python

def code(c_p, c_n, t, s):
	return math.exp(-s)

Julia

function code(c_p, c_n, t, s)
	return exp(Float64(-s))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = exp(-s);
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[Exp[(-s)], $MachinePrecision]

Derivation

1. Initial program 92.6%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
2. Add Preprocessing

3. Taylor expanded in c_p around 0

   \[\leadsto \color{blue}{c\_p \cdot \left(\left(c\_p \cdot \left(\frac{1}{2} \cdot \frac{{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} - \left(\frac{1}{2} \cdot \frac{{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{2} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} + \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot \left(\frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} - \frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}\right)\right)\right) + \frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}\right) - \frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}\right) + \frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

5. Applied rewrites 92.8%

   \[\leadsto \color{blue}{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}} \]

6. Taylor expanded in c_p around 0

   \[\leadsto 1 + \color{blue}{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)} \]

8. Applied rewrites 93.2%

   \[\leadsto e^{-s} \]
9. Add Preprocessing

Alternative 5: 3.9% accurate, 298.7× speedup

Math

\[\begin{array}{l} \\ -s \end{array} \]

FPCore

(FPCore (c_p c_n t s) :precision binary64 (- s))

C

double code(double c_p, double c_n, double t, double s) {
	return -s;
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    code = -s
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return -s;
}

Python

def code(c_p, c_n, t, s):
	return -s

Julia

function code(c_p, c_n, t, s)
	return Float64(-s)
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = -s;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := (-s)

Derivation

1. Initial program 92.6%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
2. Add Preprocessing

3. Taylor expanded in c_p around 0

   \[\leadsto \color{blue}{c\_p \cdot \left(\left(c\_p \cdot \left(\frac{1}{2} \cdot \frac{{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} - \left(\frac{1}{2} \cdot \frac{{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{2} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} + \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot \left(\frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} - \frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}\right)\right)\right) + \frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}\right) - \frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}\right) + \frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

5. Applied rewrites 92.8%

   \[\leadsto \color{blue}{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}} \]

6. Taylor expanded in c_p around 0

   \[\leadsto 1 + \color{blue}{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)} \]

8. Applied rewrites 93.2%

   \[\leadsto e^{-s} \]

9. Taylor expanded in c_p around 0

   \[\leadsto \color{blue}{c\_p \cdot \left(\left(c\_p \cdot \left(\left(\frac{1}{2} \cdot \frac{{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} + c\_p \cdot \left(\frac{1}{6} \cdot \frac{{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{3} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} - \left(\frac{1}{6} \cdot \frac{{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{3} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} + \left(\frac{1}{2} \cdot \left({\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{2} \cdot \left(\frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} - \frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}\right)\right) + \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot \left(\frac{1}{2} \cdot \frac{{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} - \left(\frac{1}{2} \cdot \frac{{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{2} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} + \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot \left(\frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} - \frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}\right)\right)\right)\right)\right)\right)\right) - \left(\frac{1}{2} \cdot \frac{{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{2} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} + \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot \left(\frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} - \frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}\right)\right)\right) + \frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}\right) - \frac{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}\right) + \frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

11. Applied rewrites 3.6%

    \[\leadsto \color{blue}{-s} \]
12. Add Preprocessing

Developer Target 1: 97.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{1 + e^{-t}}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(\frac{1 + e^{t}}{1 + e^{s}}\right)}^{c\_n} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (*
  (pow (/ (+ 1.0 (exp (- t))) (+ 1.0 (exp (- s)))) c_p)
  (pow (/ (+ 1.0 (exp t)) (+ 1.0 (exp s))) c_n)))

C

double code(double c_p, double c_n, double t, double s) {
	return pow(((1.0 + exp(-t)) / (1.0 + exp(-s))), c_p) * pow(((1.0 + exp(t)) / (1.0 + exp(s))), c_n);
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    code = (((1.0d0 + exp(-t)) / (1.0d0 + exp(-s))) ** c_p) * (((1.0d0 + exp(t)) / (1.0d0 + exp(s))) ** c_n)
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return Math.pow(((1.0 + Math.exp(-t)) / (1.0 + Math.exp(-s))), c_p) * Math.pow(((1.0 + Math.exp(t)) / (1.0 + Math.exp(s))), c_n);
}

Python

def code(c_p, c_n, t, s):
	return math.pow(((1.0 + math.exp(-t)) / (1.0 + math.exp(-s))), c_p) * math.pow(((1.0 + math.exp(t)) / (1.0 + math.exp(s))), c_n)

Julia

function code(c_p, c_n, t, s)
	return Float64((Float64(Float64(1.0 + exp(Float64(-t))) / Float64(1.0 + exp(Float64(-s)))) ^ c_p) * (Float64(Float64(1.0 + exp(t)) / Float64(1.0 + exp(s))) ^ c_n))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = (((1.0 + exp(-t)) / (1.0 + exp(-s))) ^ c_p) * (((1.0 + exp(t)) / (1.0 + exp(s))) ^ c_n);
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[(N[Power[N[(N[(1.0 + N[Exp[(-t)], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision] * N[Power[N[(N[(1.0 + N[Exp[t], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Exp[s], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024313 
(FPCore (c_p c_n t s)
  :name "Harley's example"
  :precision binary64
  :pre (and (< 0.0 c_p) (< 0.0 c_n))

  :alt
  (! :herbie-platform default (* (pow (/ (+ 1 (exp (- t))) (+ 1 (exp (- s)))) c_p) (pow (/ (+ 1 (exp t)) (+ 1 (exp s))) c_n)))

  (/ (* (pow (/ 1.0 (+ 1.0 (exp (- s)))) c_p) (pow (- 1.0 (/ 1.0 (+ 1.0 (exp (- s))))) c_n)) (* (pow (/ 1.0 (+ 1.0 (exp (- t)))) c_p) (pow (- 1.0 (/ 1.0 (+ 1.0 (exp (- t))))) c_n))))