Harley's example

Percentage Accurate: 90.8% → 98.1%

Time: 1.4min

Alternatives: 7

Speedup: 835.0×

• could not determine a ground truth

  1. c_p = 2.00000000000000007e154
  2. c_n = 2
  3. t = 0.0
  4. s = 0.0

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: 90.8% 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: 98.1% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ e^{\left(c\_p \cdot t\right) \cdot \left(-0.5\right) - s \cdot \left(c\_p \cdot -0.5 + 0.125 \cdot \left(c\_p \cdot s\right)\right)} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (exp (- (* (* c_p t) (- 0.5)) (* s (+ (* c_p -0.5) (* 0.125 (* c_p s)))))))

C

double code(double c_p, double c_n, double t, double s) {
	return exp((((c_p * t) * -0.5) - (s * ((c_p * -0.5) + (0.125 * (c_p * 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((((c_p * t) * -0.5d0) - (s * ((c_p * (-0.5d0)) + (0.125d0 * (c_p * s))))))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return Math.exp((((c_p * t) * -0.5) - (s * ((c_p * -0.5) + (0.125 * (c_p * s))))));
}

Python

def code(c_p, c_n, t, s):
	return math.exp((((c_p * t) * -0.5) - (s * ((c_p * -0.5) + (0.125 * (c_p * s))))))

Julia

function code(c_p, c_n, t, s)
	return exp(Float64(Float64(Float64(c_p * t) * Float64(-0.5)) - Float64(s * Float64(Float64(c_p * -0.5) + Float64(0.125 * Float64(c_p * s))))))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = exp((((c_p * t) * -0.5) - (s * ((c_p * -0.5) + (0.125 * (c_p * s))))));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[Exp[N[(N[(N[(c$95$p * t), $MachinePrecision] * (-0.5)), $MachinePrecision] - N[(s * N[(N[(c$95$p * -0.5), $MachinePrecision] + N[(0.125 * N[(c$95$p * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 88.4%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
2. Step-by-step derivation

   1. associate-/l/ 88.4%

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

3. Simplified 88.4%

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

5. Taylor expanded in c_n around 0 91.0%

   \[\leadsto \frac{\color{blue}{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p}} \]
6. Step-by-step derivation

   1. clear-num 91.0%

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

   2. inv-pow 91.0%

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

7. Applied egg-rr 93.1%

   \[\leadsto \color{blue}{{\left(e^{c\_p \cdot \left(-\mathsf{log1p}\left(e^{-t}\right)\right) - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-s}\right)\right)}\right)}^{-1}} \]
8. Step-by-step derivation

   1. unpow-1 93.1%

      \[\leadsto \color{blue}{\frac{1}{e^{c\_p \cdot \left(-\mathsf{log1p}\left(e^{-t}\right)\right) - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-s}\right)\right)}}} \]

   2. rec-exp 93.1%

      \[\leadsto \color{blue}{e^{-\left(c\_p \cdot \left(-\mathsf{log1p}\left(e^{-t}\right)\right) - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-s}\right)\right)\right)}} \]

   3. log1p-undefine 93.1%

      \[\leadsto e^{-\left(c\_p \cdot \left(-\color{blue}{\log \left(1 + e^{-t}\right)}\right) - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-s}\right)\right)\right)} \]

   4. log-rec 93.1%

      \[\leadsto e^{-\left(c\_p \cdot \color{blue}{\log \left(\frac{1}{1 + e^{-t}}\right)} - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-s}\right)\right)\right)} \]

   5. log1p-undefine 93.1%

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

   6. log-rec 93.1%

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

   7. distribute-lft-out-- 93.1%

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

   8. log-rec 93.1%

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

   9. log1p-undefine 93.1%

      \[\leadsto e^{-c\_p \cdot \left(\left(-\color{blue}{\mathsf{log1p}\left(e^{-t}\right)}\right) - \log \left(\frac{1}{1 + e^{-s}}\right)\right)} \]

   10. neg-mul-1 93.1%

       \[\leadsto e^{-c\_p \cdot \left(\color{blue}{-1 \cdot \mathsf{log1p}\left(e^{-t}\right)} - \log \left(\frac{1}{1 + e^{-s}}\right)\right)} \]

9. Simplified 93.1%

   \[\leadsto \color{blue}{e^{-c\_p \cdot \left(-1 \cdot \left(\mathsf{log1p}\left(e^{-t}\right) - \mathsf{log1p}\left(e^{-s}\right)\right)\right)}} \]

10. Taylor expanded in t around 0 94.1%

    \[\leadsto e^{-\color{blue}{\left(-1 \cdot \left(c\_p \cdot \left(\log 2 - \log \left(1 + e^{-s}\right)\right)\right) + 0.5 \cdot \left(c\_p \cdot t\right)\right)}} \]

11. Taylor expanded in s around 0 98.6%

    \[\leadsto e^{-\color{blue}{\left(0.5 \cdot \left(c\_p \cdot t\right) + s \cdot \left(-0.5 \cdot c\_p + 0.125 \cdot \left(c\_p \cdot s\right)\right)\right)}} \]

12. Final simplification 98.6%

    \[\leadsto e^{\left(c\_p \cdot t\right) \cdot \left(-0.5\right) - s \cdot \left(c\_p \cdot -0.5 + 0.125 \cdot \left(c\_p \cdot s\right)\right)} \]
13. Add Preprocessing

Alternative 2: 97.5% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c\_n \leq 100:\\ \;\;\;\;e^{\left(c\_p \cdot t\right) \cdot \left(-0.5\right) - -0.5 \cdot \left(c\_p \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(0.5 \cdot s + -1\right)}\right)}^{c\_p}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= c_n 100.0)
   (exp (- (* (* c_p t) (- 0.5)) (* -0.5 (* c_p s))))
   (pow (/ 1.0 (+ 2.0 (* s (+ (* 0.5 s) -1.0)))) c_p)))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_n <= 100.0) {
		tmp = exp((((c_p * t) * -0.5) - (-0.5 * (c_p * s))));
	} else {
		tmp = pow((1.0 / (2.0 + (s * ((0.5 * s) + -1.0)))), 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) :: tmp
    if (c_n <= 100.0d0) then
        tmp = exp((((c_p * t) * -0.5d0) - ((-0.5d0) * (c_p * s))))
    else
        tmp = (1.0d0 / (2.0d0 + (s * ((0.5d0 * s) + (-1.0d0))))) ** c_p
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_n <= 100.0) {
		tmp = Math.exp((((c_p * t) * -0.5) - (-0.5 * (c_p * s))));
	} else {
		tmp = Math.pow((1.0 / (2.0 + (s * ((0.5 * s) + -1.0)))), c_p);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if c_n <= 100.0:
		tmp = math.exp((((c_p * t) * -0.5) - (-0.5 * (c_p * s))))
	else:
		tmp = math.pow((1.0 / (2.0 + (s * ((0.5 * s) + -1.0)))), c_p)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (c_n <= 100.0)
		tmp = exp(Float64(Float64(Float64(c_p * t) * Float64(-0.5)) - Float64(-0.5 * Float64(c_p * s))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(s * Float64(Float64(0.5 * s) + -1.0)))) ^ c_p;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (c_n <= 100.0)
		tmp = exp((((c_p * t) * -0.5) - (-0.5 * (c_p * s))));
	else
		tmp = (1.0 / (2.0 + (s * ((0.5 * s) + -1.0)))) ^ c_p;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[c$95$n, 100.0], N[Exp[N[(N[(N[(c$95$p * t), $MachinePrecision] * (-0.5)), $MachinePrecision] - N[(-0.5 * N[(c$95$p * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(1.0 / N[(2.0 + N[(s * N[(N[(0.5 * s), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c_n < 100

   1. Initial program 95.0%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. Step-by-step derivation

      1. associate-/l/ 95.0%

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

   3. Simplified 95.0%

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

   5. Taylor expanded in c_n around 0 95.6%

      \[\leadsto \frac{\color{blue}{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p}} \]
   6. Step-by-step derivation

      1. clear-num 95.6%

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

      2. inv-pow 95.6%

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

   7. Applied egg-rr 97.5%

      \[\leadsto \color{blue}{{\left(e^{c\_p \cdot \left(-\mathsf{log1p}\left(e^{-t}\right)\right) - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-s}\right)\right)}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 97.5%

         \[\leadsto \color{blue}{\frac{1}{e^{c\_p \cdot \left(-\mathsf{log1p}\left(e^{-t}\right)\right) - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-s}\right)\right)}}} \]

      2. rec-exp 97.5%

         \[\leadsto \color{blue}{e^{-\left(c\_p \cdot \left(-\mathsf{log1p}\left(e^{-t}\right)\right) - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-s}\right)\right)\right)}} \]

      3. log1p-undefine 97.5%

         \[\leadsto e^{-\left(c\_p \cdot \left(-\color{blue}{\log \left(1 + e^{-t}\right)}\right) - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-s}\right)\right)\right)} \]

      4. log-rec 97.5%

         \[\leadsto e^{-\left(c\_p \cdot \color{blue}{\log \left(\frac{1}{1 + e^{-t}}\right)} - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-s}\right)\right)\right)} \]

      5. log1p-undefine 97.5%

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

      6. log-rec 97.5%

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

      7. distribute-lft-out-- 97.5%

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

      8. log-rec 97.5%

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

      9. log1p-undefine 97.5%

         \[\leadsto e^{-c\_p \cdot \left(\left(-\color{blue}{\mathsf{log1p}\left(e^{-t}\right)}\right) - \log \left(\frac{1}{1 + e^{-s}}\right)\right)} \]

      10. neg-mul-1 97.5%

          \[\leadsto e^{-c\_p \cdot \left(\color{blue}{-1 \cdot \mathsf{log1p}\left(e^{-t}\right)} - \log \left(\frac{1}{1 + e^{-s}}\right)\right)} \]

   9. Simplified 97.5%

      \[\leadsto \color{blue}{e^{-c\_p \cdot \left(-1 \cdot \left(\mathsf{log1p}\left(e^{-t}\right) - \mathsf{log1p}\left(e^{-s}\right)\right)\right)}} \]

   10. Taylor expanded in t around 0 98.5%

       \[\leadsto e^{-\color{blue}{\left(-1 \cdot \left(c\_p \cdot \left(\log 2 - \log \left(1 + e^{-s}\right)\right)\right) + 0.5 \cdot \left(c\_p \cdot t\right)\right)}} \]

   11. Taylor expanded in s around 0 99.3%

       \[\leadsto e^{-\color{blue}{\left(-0.5 \cdot \left(c\_p \cdot s\right) + 0.5 \cdot \left(c\_p \cdot t\right)\right)}} \]

   if 100 < c_n

   1. Initial program 5.3%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. Step-by-step derivation

      1. associate-/l/ 5.3%

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

   3. Simplified 5.3%

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

   5. Taylor expanded in c_n around 0 33.1%

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

   6. Taylor expanded in c_p around 0 38.8%

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

   7. Taylor expanded in s around 0 74.5%

      \[\leadsto \frac{{\left(\frac{1}{\color{blue}{2 + s \cdot \left(0.5 \cdot s - 1\right)}}\right)}^{c\_p}}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_n \leq 100:\\ \;\;\;\;e^{\left(c\_p \cdot t\right) \cdot \left(-0.5\right) - -0.5 \cdot \left(c\_p \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(0.5 \cdot s + -1\right)}\right)}^{c\_p}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.8% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ e^{\left(c\_p \cdot t\right) \cdot \left(-0.5\right) - -0.5 \cdot \left(c\_p \cdot s\right)} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (exp (- (* (* c_p t) (- 0.5)) (* -0.5 (* c_p s)))))

C

double code(double c_p, double c_n, double t, double s) {
	return exp((((c_p * t) * -0.5) - (-0.5 * (c_p * 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((((c_p * t) * -0.5d0) - ((-0.5d0) * (c_p * s))))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return Math.exp((((c_p * t) * -0.5) - (-0.5 * (c_p * s))));
}

Python

def code(c_p, c_n, t, s):
	return math.exp((((c_p * t) * -0.5) - (-0.5 * (c_p * s))))

Julia

function code(c_p, c_n, t, s)
	return exp(Float64(Float64(Float64(c_p * t) * Float64(-0.5)) - Float64(-0.5 * Float64(c_p * s))))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = exp((((c_p * t) * -0.5) - (-0.5 * (c_p * s))));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[Exp[N[(N[(N[(c$95$p * t), $MachinePrecision] * (-0.5)), $MachinePrecision] - N[(-0.5 * N[(c$95$p * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 88.4%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
2. Step-by-step derivation

   1. associate-/l/ 88.4%

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

3. Simplified 88.4%

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

5. Taylor expanded in c_n around 0 91.0%

   \[\leadsto \frac{\color{blue}{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p}} \]
6. Step-by-step derivation

   1. clear-num 91.0%

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

   2. inv-pow 91.0%

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

7. Applied egg-rr 93.1%

   \[\leadsto \color{blue}{{\left(e^{c\_p \cdot \left(-\mathsf{log1p}\left(e^{-t}\right)\right) - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-s}\right)\right)}\right)}^{-1}} \]
8. Step-by-step derivation

   1. unpow-1 93.1%

      \[\leadsto \color{blue}{\frac{1}{e^{c\_p \cdot \left(-\mathsf{log1p}\left(e^{-t}\right)\right) - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-s}\right)\right)}}} \]

   2. rec-exp 93.1%

      \[\leadsto \color{blue}{e^{-\left(c\_p \cdot \left(-\mathsf{log1p}\left(e^{-t}\right)\right) - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-s}\right)\right)\right)}} \]

   3. log1p-undefine 93.1%

      \[\leadsto e^{-\left(c\_p \cdot \left(-\color{blue}{\log \left(1 + e^{-t}\right)}\right) - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-s}\right)\right)\right)} \]

   4. log-rec 93.1%

      \[\leadsto e^{-\left(c\_p \cdot \color{blue}{\log \left(\frac{1}{1 + e^{-t}}\right)} - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-s}\right)\right)\right)} \]

   5. log1p-undefine 93.1%

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

   6. log-rec 93.1%

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

   7. distribute-lft-out-- 93.1%

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

   8. log-rec 93.1%

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

   9. log1p-undefine 93.1%

      \[\leadsto e^{-c\_p \cdot \left(\left(-\color{blue}{\mathsf{log1p}\left(e^{-t}\right)}\right) - \log \left(\frac{1}{1 + e^{-s}}\right)\right)} \]

   10. neg-mul-1 93.1%

       \[\leadsto e^{-c\_p \cdot \left(\color{blue}{-1 \cdot \mathsf{log1p}\left(e^{-t}\right)} - \log \left(\frac{1}{1 + e^{-s}}\right)\right)} \]

9. Simplified 93.1%

   \[\leadsto \color{blue}{e^{-c\_p \cdot \left(-1 \cdot \left(\mathsf{log1p}\left(e^{-t}\right) - \mathsf{log1p}\left(e^{-s}\right)\right)\right)}} \]

10. Taylor expanded in t around 0 94.1%

    \[\leadsto e^{-\color{blue}{\left(-1 \cdot \left(c\_p \cdot \left(\log 2 - \log \left(1 + e^{-s}\right)\right)\right) + 0.5 \cdot \left(c\_p \cdot t\right)\right)}} \]

11. Taylor expanded in s around 0 94.8%

    \[\leadsto e^{-\color{blue}{\left(-0.5 \cdot \left(c\_p \cdot s\right) + 0.5 \cdot \left(c\_p \cdot t\right)\right)}} \]

12. Final simplification 94.8%

    \[\leadsto e^{\left(c\_p \cdot t\right) \cdot \left(-0.5\right) - -0.5 \cdot \left(c\_p \cdot s\right)} \]
13. Add Preprocessing

Alternative 4: 94.2% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ e^{t \cdot \left(c\_p \cdot \left(-0.5\right)\right)} \end{array} \]

FPCore

(FPCore (c_p c_n t s) :precision binary64 (exp (* t (* c_p (- 0.5)))))

C

double code(double c_p, double c_n, double t, double s) {
	return exp((t * (c_p * -0.5)));
}

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((t * (c_p * -0.5d0)))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return Math.exp((t * (c_p * -0.5)));
}

Python

def code(c_p, c_n, t, s):
	return math.exp((t * (c_p * -0.5)))

Julia

function code(c_p, c_n, t, s)
	return exp(Float64(t * Float64(c_p * Float64(-0.5))))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = exp((t * (c_p * -0.5)));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[Exp[N[(t * N[(c$95$p * (-0.5)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 88.4%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
2. Step-by-step derivation

   1. associate-/l/ 88.4%

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

3. Simplified 88.4%

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

5. Taylor expanded in c_n around 0 91.0%

   \[\leadsto \frac{\color{blue}{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p}} \]
6. Step-by-step derivation

   1. clear-num 91.0%

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

   2. inv-pow 91.0%

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

7. Applied egg-rr 93.1%

   \[\leadsto \color{blue}{{\left(e^{c\_p \cdot \left(-\mathsf{log1p}\left(e^{-t}\right)\right) - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-s}\right)\right)}\right)}^{-1}} \]
8. Step-by-step derivation

   1. unpow-1 93.1%

      \[\leadsto \color{blue}{\frac{1}{e^{c\_p \cdot \left(-\mathsf{log1p}\left(e^{-t}\right)\right) - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-s}\right)\right)}}} \]

   2. rec-exp 93.1%

      \[\leadsto \color{blue}{e^{-\left(c\_p \cdot \left(-\mathsf{log1p}\left(e^{-t}\right)\right) - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-s}\right)\right)\right)}} \]

   3. log1p-undefine 93.1%

      \[\leadsto e^{-\left(c\_p \cdot \left(-\color{blue}{\log \left(1 + e^{-t}\right)}\right) - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-s}\right)\right)\right)} \]

   4. log-rec 93.1%

      \[\leadsto e^{-\left(c\_p \cdot \color{blue}{\log \left(\frac{1}{1 + e^{-t}}\right)} - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-s}\right)\right)\right)} \]

   5. log1p-undefine 93.1%

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

   6. log-rec 93.1%

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

   7. distribute-lft-out-- 93.1%

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

   8. log-rec 93.1%

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

   9. log1p-undefine 93.1%

      \[\leadsto e^{-c\_p \cdot \left(\left(-\color{blue}{\mathsf{log1p}\left(e^{-t}\right)}\right) - \log \left(\frac{1}{1 + e^{-s}}\right)\right)} \]

   10. neg-mul-1 93.1%

       \[\leadsto e^{-c\_p \cdot \left(\color{blue}{-1 \cdot \mathsf{log1p}\left(e^{-t}\right)} - \log \left(\frac{1}{1 + e^{-s}}\right)\right)} \]

9. Simplified 93.1%

   \[\leadsto \color{blue}{e^{-c\_p \cdot \left(-1 \cdot \left(\mathsf{log1p}\left(e^{-t}\right) - \mathsf{log1p}\left(e^{-s}\right)\right)\right)}} \]

10. Taylor expanded in t around 0 94.1%

    \[\leadsto e^{-\color{blue}{\left(-1 \cdot \left(c\_p \cdot \left(\log 2 - \log \left(1 + e^{-s}\right)\right)\right) + 0.5 \cdot \left(c\_p \cdot t\right)\right)}} \]

11. Taylor expanded in s around 0 93.3%

    \[\leadsto e^{-\color{blue}{0.5 \cdot \left(c\_p \cdot t\right)}} \]
12. Step-by-step derivation

    1. associate-*r* 93.3%

       \[\leadsto e^{-\color{blue}{\left(0.5 \cdot c\_p\right) \cdot t}} \]

    2. *-commutative 93.3%

       \[\leadsto e^{-\color{blue}{t \cdot \left(0.5 \cdot c\_p\right)}} \]

13. Simplified 93.3%

    \[\leadsto e^{-\color{blue}{t \cdot \left(0.5 \cdot c\_p\right)}} \]

14. Final simplification 93.3%

    \[\leadsto e^{t \cdot \left(c\_p \cdot \left(-0.5\right)\right)} \]
15. Add Preprocessing

Alternative 5: 93.9% accurate, 75.9× speedup

Math

\[\begin{array}{l} \\ 1 - t \cdot \left(0.5 \cdot c\_p + -0.5 \cdot c\_n\right) \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (- 1.0 (* t (+ (* 0.5 c_p) (* -0.5 c_n)))))

C

double code(double c_p, double c_n, double t, double s) {
	return 1.0 - (t * ((0.5 * c_p) + (-0.5 * 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 - (t * ((0.5d0 * c_p) + ((-0.5d0) * c_n)))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return 1.0 - (t * ((0.5 * c_p) + (-0.5 * c_n)));
}

Python

def code(c_p, c_n, t, s):
	return 1.0 - (t * ((0.5 * c_p) + (-0.5 * c_n)))

Julia

function code(c_p, c_n, t, s)
	return Float64(1.0 - Float64(t * Float64(Float64(0.5 * c_p) + Float64(-0.5 * c_n))))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = 1.0 - (t * ((0.5 * c_p) + (-0.5 * c_n)));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[(1.0 - N[(t * N[(N[(0.5 * c$95$p), $MachinePrecision] + N[(-0.5 * c$95$n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.4%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
2. Step-by-step derivation

   1. associate-/l/ 88.4%

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

3. Simplified 88.4%

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

5. Taylor expanded in s around 0 88.0%

   \[\leadsto \frac{\color{blue}{\frac{{0.5}^{c\_n} \cdot {0.5}^{c\_p}}{{\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}}}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p}} \]

6. Taylor expanded in t around 0 92.7%

   \[\leadsto \color{blue}{1 + -1 \cdot \left(t \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right)\right)} \]

7. Final simplification 92.7%

   \[\leadsto 1 - t \cdot \left(0.5 \cdot c\_p + -0.5 \cdot c\_n\right) \]
8. Add Preprocessing

Alternative 6: 93.9% accurate, 119.3× speedup

Math

\[\begin{array}{l} \\ 1 - t \cdot \left(-0.5 \cdot c\_n\right) \end{array} \]

FPCore

(FPCore (c_p c_n t s) :precision binary64 (- 1.0 (* t (* -0.5 c_n))))

C

double code(double c_p, double c_n, double t, double s) {
	return 1.0 - (t * (-0.5 * 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 - (t * ((-0.5d0) * c_n))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return 1.0 - (t * (-0.5 * c_n));
}

Python

def code(c_p, c_n, t, s):
	return 1.0 - (t * (-0.5 * c_n))

Julia

function code(c_p, c_n, t, s)
	return Float64(1.0 - Float64(t * Float64(-0.5 * c_n)))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = 1.0 - (t * (-0.5 * c_n));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[(1.0 - N[(t * N[(-0.5 * c$95$n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.4%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
2. Step-by-step derivation

   1. associate-/l/ 88.4%

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

3. Simplified 88.4%

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

5. Taylor expanded in s around 0 88.0%

   \[\leadsto \frac{\color{blue}{\frac{{0.5}^{c\_n} \cdot {0.5}^{c\_p}}{{\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}}}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p}} \]

6. Taylor expanded in t around 0 92.7%

   \[\leadsto \color{blue}{1 + -1 \cdot \left(t \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right)\right)} \]

7. Taylor expanded in c_n around inf 92.6%

   \[\leadsto 1 + -1 \cdot \color{blue}{\left(-0.5 \cdot \left(c\_n \cdot t\right)\right)} \]
8. Step-by-step derivation

   1. associate-*r* 92.6%

      \[\leadsto 1 + -1 \cdot \color{blue}{\left(\left(-0.5 \cdot c\_n\right) \cdot t\right)} \]

   2. *-commutative 92.6%

      \[\leadsto 1 + -1 \cdot \color{blue}{\left(t \cdot \left(-0.5 \cdot c\_n\right)\right)} \]

9. Simplified 92.6%

   \[\leadsto 1 + -1 \cdot \color{blue}{\left(t \cdot \left(-0.5 \cdot c\_n\right)\right)} \]

10. Final simplification 92.6%

    \[\leadsto 1 - t \cdot \left(-0.5 \cdot c\_n\right) \]
11. Add Preprocessing

Alternative 7: 93.8% accurate, 835.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (c_p c_n t s) :precision binary64 1.0)

C

double code(double c_p, double c_n, double t, double s) {
	return 1.0;
}

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
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return 1.0;
}

Python

def code(c_p, c_n, t, s):
	return 1.0

Julia

function code(c_p, c_n, t, s)
	return 1.0
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = 1.0;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := 1.0

Derivation

1. Initial program 88.4%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
2. Step-by-step derivation

   1. associate-/l/ 88.4%

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

3. Simplified 88.4%

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

5. Taylor expanded in c_n around 0 91.0%

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

6. Taylor expanded in c_p around 0 92.5%

   \[\leadsto \color{blue}{1} \]
7. Add Preprocessing

Developer target: 96.5% accurate, 1.3× 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 2024101 
(FPCore (c_p c_n t s)
  :name "Harley's example"
  :precision binary64
  :pre (and (< 0.0 c_p) (< 0.0 c_n))

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