Harley's example

Percentage Accurate: 91.3% → 98.6%

Time: 1.4min

Alternatives: 5

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: 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: 98.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c\_p \leq 0.00038:\\ \;\;\;\;e^{c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{1 + e^{s}}\right) - \mathsf{log1p}\left(\frac{1}{1 + e^{t}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{c\_p \cdot \left(\mathsf{log1p}\left(e^{-t}\right) - \mathsf{log1p}\left(e^{-s}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= c_p 0.00038)
   (exp
    (*
     c_n
     (- (log1p (/ 1.0 (+ 1.0 (exp s)))) (log1p (/ 1.0 (+ 1.0 (exp t)))))))
   (exp (* c_p (- (log1p (exp (- t))) (log1p (exp (- s))))))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 0.00038) {
		tmp = exp((c_n * (log1p((1.0 / (1.0 + exp(s)))) - log1p((1.0 / (1.0 + exp(t)))))));
	} else {
		tmp = exp((c_p * (log1p(exp(-t)) - log1p(exp(-s)))));
	}
	return tmp;
}

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 0.00038) {
		tmp = Math.exp((c_n * (Math.log1p((1.0 / (1.0 + Math.exp(s)))) - Math.log1p((1.0 / (1.0 + Math.exp(t)))))));
	} else {
		tmp = Math.exp((c_p * (Math.log1p(Math.exp(-t)) - Math.log1p(Math.exp(-s)))));
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if c_p <= 0.00038:
		tmp = math.exp((c_n * (math.log1p((1.0 / (1.0 + math.exp(s)))) - math.log1p((1.0 / (1.0 + math.exp(t)))))))
	else:
		tmp = math.exp((c_p * (math.log1p(math.exp(-t)) - math.log1p(math.exp(-s)))))
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (c_p <= 0.00038)
		tmp = exp(Float64(c_n * Float64(log1p(Float64(1.0 / Float64(1.0 + exp(s)))) - log1p(Float64(1.0 / Float64(1.0 + exp(t)))))));
	else
		tmp = exp(Float64(c_p * Float64(log1p(exp(Float64(-t))) - log1p(exp(Float64(-s))))));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[c$95$p, 0.00038], N[Exp[N[(c$95$n * N[(N[Log[1 + N[(1.0 / N[(1.0 + N[Exp[s], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[1 + N[(1.0 / N[(1.0 + N[Exp[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(c$95$p * N[(N[Log[1 + N[Exp[(-t)], $MachinePrecision]], $MachinePrecision] - N[Log[1 + N[Exp[(-s)], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c_p < 3.8000000000000002e-4

   1. Initial program 91.3%

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

      1. associate-/l/ 91.3%

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

   3. Simplified 91.3%

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

   6. Applied egg-rr 99.6%

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

      1. *-lft-identity 99.6%

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

      2. associate--l+ 99.6%

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

      3. distribute-lft-out-- 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in c_p around 0 100.0%

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

       1. log1p-define 100.0%

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

       2. log1p-define 100.0%

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

   11. Simplified 100.0%

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

   if 3.8000000000000002e-4 < c_p

   1. Initial program 68.5%

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

      1. associate-/l/ 68.5%

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

   3. Simplified 68.5%

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

   5. Taylor expanded in c_n around 0 68.5%

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

      1. clear-num 68.5%

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

      2. inv-pow 68.5%

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

   7. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

      2. rec-exp 99.7%

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

      3. log1p-undefine 99.6%

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

      4. log-rec 99.6%

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

      5. log1p-undefine 99.6%

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

      6. log-rec 99.6%

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

      7. distribute-lft-out-- 99.7%

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

      8. log-rec 99.7%

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

      9. log1p-undefine 100.0%

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

      10. log-rec 100.0%

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

   9. Simplified 100.0%

      \[\leadsto \color{blue}{e^{-c\_p \cdot \left(\left(-\mathsf{log1p}\left(e^{-t}\right)\right) - \left(-\mathsf{log1p}\left(e^{-s}\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_p \leq 0.00038:\\ \;\;\;\;e^{c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{1 + e^{s}}\right) - \mathsf{log1p}\left(\frac{1}{1 + e^{t}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{c\_p \cdot \left(\mathsf{log1p}\left(e^{-t}\right) - \mathsf{log1p}\left(e^{-s}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c\_p \leq 0.00037:\\ \;\;\;\;e^{c\_n \cdot \left(t \cdot 0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{c\_p \cdot \left(\mathsf{log1p}\left(e^{-t}\right) - \mathsf{log1p}\left(e^{-s}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= c_p 0.00037)
   (exp (* c_n (* t 0.16666666666666666)))
   (exp (* c_p (- (log1p (exp (- t))) (log1p (exp (- s))))))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 0.00037) {
		tmp = exp((c_n * (t * 0.16666666666666666)));
	} else {
		tmp = exp((c_p * (log1p(exp(-t)) - log1p(exp(-s)))));
	}
	return tmp;
}

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 0.00037) {
		tmp = Math.exp((c_n * (t * 0.16666666666666666)));
	} else {
		tmp = Math.exp((c_p * (Math.log1p(Math.exp(-t)) - Math.log1p(Math.exp(-s)))));
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if c_p <= 0.00037:
		tmp = math.exp((c_n * (t * 0.16666666666666666)))
	else:
		tmp = math.exp((c_p * (math.log1p(math.exp(-t)) - math.log1p(math.exp(-s)))))
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (c_p <= 0.00037)
		tmp = exp(Float64(c_n * Float64(t * 0.16666666666666666)));
	else
		tmp = exp(Float64(c_p * Float64(log1p(exp(Float64(-t))) - log1p(exp(Float64(-s))))));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[c$95$p, 0.00037], N[Exp[N[(c$95$n * N[(t * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(c$95$p * N[(N[Log[1 + N[Exp[(-t)], $MachinePrecision]], $MachinePrecision] - N[Log[1 + N[Exp[(-s)], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c_p < 3.6999999999999999e-4

   1. Initial program 91.3%

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

      1. associate-/l/ 91.3%

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

   3. Simplified 91.3%

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

   6. Applied egg-rr 99.6%

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

      1. *-lft-identity 99.6%

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

      2. associate--l+ 99.6%

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

      3. distribute-lft-out-- 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in c_p around 0 100.0%

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

       1. log1p-define 100.0%

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

       2. log1p-define 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in s around 0 98.8%

       \[\leadsto e^{\color{blue}{c\_n \cdot \left(\log 1.5 - \log \left(1 + \frac{1}{1 + e^{t}}\right)\right)}} \]
   13. Step-by-step derivation

       1. log1p-define 98.8%

          \[\leadsto e^{c\_n \cdot \left(\log 1.5 - \color{blue}{\mathsf{log1p}\left(\frac{1}{1 + e^{t}}\right)}\right)} \]

   14. Simplified 98.8%

       \[\leadsto e^{\color{blue}{c\_n \cdot \left(\log 1.5 - \mathsf{log1p}\left(\frac{1}{1 + e^{t}}\right)\right)}} \]

   15. Taylor expanded in t around 0 98.8%

       \[\leadsto e^{c\_n \cdot \color{blue}{\left(0.16666666666666666 \cdot t\right)}} \]

   if 3.6999999999999999e-4 < c_p

   1. Initial program 68.5%

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

      1. associate-/l/ 68.5%

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

   3. Simplified 68.5%

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

   5. Taylor expanded in c_n around 0 68.5%

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

      1. clear-num 68.5%

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

      2. inv-pow 68.5%

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

   7. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

      2. rec-exp 99.7%

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

      3. log1p-undefine 99.6%

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

      4. log-rec 99.6%

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

      5. log1p-undefine 99.6%

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

      6. log-rec 99.6%

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

      7. distribute-lft-out-- 99.7%

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

      8. log-rec 99.7%

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

      9. log1p-undefine 100.0%

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

      10. log-rec 100.0%

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

   9. Simplified 100.0%

      \[\leadsto \color{blue}{e^{-c\_p \cdot \left(\left(-\mathsf{log1p}\left(e^{-t}\right)\right) - \left(-\mathsf{log1p}\left(e^{-s}\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_p \leq 0.00037:\\ \;\;\;\;e^{c\_n \cdot \left(t \cdot 0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{c\_p \cdot \left(\mathsf{log1p}\left(e^{-t}\right) - \mathsf{log1p}\left(e^{-s}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.8% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ e^{c\_n \cdot \left(t \cdot 0.16666666666666666\right)} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (exp (* c_n (* t 0.16666666666666666))))

C

double code(double c_p, double c_n, double t, double s) {
	return exp((c_n * (t * 0.16666666666666666)));
}

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_n * (t * 0.16666666666666666d0)))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return Math.exp((c_n * (t * 0.16666666666666666)));
}

Python

def code(c_p, c_n, t, s):
	return math.exp((c_n * (t * 0.16666666666666666)))

Julia

function code(c_p, c_n, t, s)
	return exp(Float64(c_n * Float64(t * 0.16666666666666666)))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = exp((c_n * (t * 0.16666666666666666)));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[Exp[N[(c$95$n * N[(t * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 89.9%

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

   1. associate-/l/ 89.9%

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

3. Simplified 89.9%

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

6. Applied egg-rr 97.8%

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

   1. *-lft-identity 97.8%

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

   2. associate--l+ 97.8%

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

   3. distribute-lft-out-- 97.8%

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

8. Simplified 97.8%

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

9. Taylor expanded in c_p around 0 98.2%

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

    1. log1p-define 98.2%

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

    2. log1p-define 98.2%

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

11. Simplified 98.2%

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

12. Taylor expanded in s around 0 97.1%

    \[\leadsto e^{\color{blue}{c\_n \cdot \left(\log 1.5 - \log \left(1 + \frac{1}{1 + e^{t}}\right)\right)}} \]
13. Step-by-step derivation

    1. log1p-define 97.1%

       \[\leadsto e^{c\_n \cdot \left(\log 1.5 - \color{blue}{\mathsf{log1p}\left(\frac{1}{1 + e^{t}}\right)}\right)} \]

14. Simplified 97.1%

    \[\leadsto e^{\color{blue}{c\_n \cdot \left(\log 1.5 - \mathsf{log1p}\left(\frac{1}{1 + e^{t}}\right)\right)}} \]

15. Taylor expanded in t around 0 97.1%

    \[\leadsto e^{c\_n \cdot \color{blue}{\left(0.16666666666666666 \cdot t\right)}} \]

16. Final simplification 97.1%

    \[\leadsto e^{c\_n \cdot \left(t \cdot 0.16666666666666666\right)} \]
17. Add Preprocessing

Alternative 4: 94.6% accurate, 119.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.9%

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

   1. associate-/l/ 89.9%

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

3. Simplified 89.9%

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

5. Taylor expanded in s around 0 91.0%

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

6. Taylor expanded in t around 0 94.9%

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

7. Taylor expanded in c_n around 0 94.9%

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

   1. associate-*r* 94.9%

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

   2. *-commutative 94.9%

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

9. Simplified 94.9%

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

10. Final simplification 94.9%

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

Alternative 5: 94.6% 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 89.9%

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

   1. associate-/l/ 89.9%

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

3. Simplified 89.9%

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

5. Taylor expanded in c_n around 0 92.7%

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

6. Taylor expanded in c_p around 0 94.8%

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

Developer target: 96.7% 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 2024103 
(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))))