Harley's example

Percentage Accurate: 90.7% → 99.1%

Time: 2.4min

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: 90.7% 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: 99.1% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{e^{t \cdot \left(-0.5 \cdot c\_n + -0.5 \cdot c\_p\right) - s \cdot \left(-0.5 \cdot c\_n + \left(c\_p \cdot 0.5 + s \cdot \left(c\_n \cdot -0.125 + c\_p \cdot -0.125\right)\right)\right)}} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (/
  1.0
  (exp
   (-
    (* t (+ (* -0.5 c_n) (* -0.5 c_p)))
    (*
     s
     (+
      (* -0.5 c_n)
      (+ (* c_p 0.5) (* s (+ (* c_n -0.125) (* c_p -0.125))))))))))

C

double code(double c_p, double c_n, double t, double s) {
	return 1.0 / exp(((t * ((-0.5 * c_n) + (-0.5 * c_p))) - (s * ((-0.5 * c_n) + ((c_p * 0.5) + (s * ((c_n * -0.125) + (c_p * -0.125))))))));
}

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 * (((-0.5d0) * c_n) + ((-0.5d0) * c_p))) - (s * (((-0.5d0) * c_n) + ((c_p * 0.5d0) + (s * ((c_n * (-0.125d0)) + (c_p * (-0.125d0)))))))))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return 1.0 / Math.exp(((t * ((-0.5 * c_n) + (-0.5 * c_p))) - (s * ((-0.5 * c_n) + ((c_p * 0.5) + (s * ((c_n * -0.125) + (c_p * -0.125))))))));
}

Python

def code(c_p, c_n, t, s):
	return 1.0 / math.exp(((t * ((-0.5 * c_n) + (-0.5 * c_p))) - (s * ((-0.5 * c_n) + ((c_p * 0.5) + (s * ((c_n * -0.125) + (c_p * -0.125))))))))

Julia

function code(c_p, c_n, t, s)
	return Float64(1.0 / exp(Float64(Float64(t * Float64(Float64(-0.5 * c_n) + Float64(-0.5 * c_p))) - Float64(s * Float64(Float64(-0.5 * c_n) + Float64(Float64(c_p * 0.5) + Float64(s * Float64(Float64(c_n * -0.125) + Float64(c_p * -0.125)))))))))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = 1.0 / exp(((t * ((-0.5 * c_n) + (-0.5 * c_p))) - (s * ((-0.5 * c_n) + ((c_p * 0.5) + (s * ((c_n * -0.125) + (c_p * -0.125))))))));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[(1.0 / N[Exp[N[(N[(t * N[(N[(-0.5 * c$95$n), $MachinePrecision] + N[(-0.5 * c$95$p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(s * N[(N[(-0.5 * c$95$n), $MachinePrecision] + N[(N[(c$95$p * 0.5), $MachinePrecision] + N[(s * N[(N[(c$95$n * -0.125), $MachinePrecision] + N[(c$95$p * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.3%

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

   1. associate-/l/ 88.3%

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

3. Simplified 88.3%

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

   1. add-log-exp 87.9%

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

   2. inv-pow 87.9%

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

   3. pow-pow 87.9%

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

   4. add-sqr-sqrt 42.2%

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

   5. sqrt-unprod 87.9%

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

   6. sqr-neg 87.9%

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

   7. sqrt-unprod 45.7%

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

   8. add-sqr-sqrt 90.6%

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

6. Applied egg-rr 90.6%

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

8. Applied egg-rr 97.5%

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

   1. unpow-1 97.5%

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

10. Simplified 97.4%

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

11. Taylor expanded in s around 0 98.9%

    \[\leadsto \frac{1}{e^{\left(\mathsf{log1p}\left(e^{t}\right) \cdot \left(-c\_p\right) - \color{blue}{\left(-1 \cdot \left(c\_p \cdot \log 2\right) + \left(c\_n \cdot \log 0.5 + s \cdot \left(-0.5 \cdot c\_n + \left(0.5 \cdot c\_p + s \cdot \left(-0.125 \cdot c\_n + -0.125 \cdot c\_p\right)\right)\right)\right)\right)}\right) + c\_n \cdot \mathsf{log1p}\left(\frac{1}{-1 - e^{-t}}\right)}} \]

12. Taylor expanded in t around 0 99.6%

    \[\leadsto \frac{1}{e^{\color{blue}{t \cdot \left(-0.5 \cdot c\_n + -0.5 \cdot c\_p\right) - s \cdot \left(-0.5 \cdot c\_n + \left(0.5 \cdot c\_p + s \cdot \left(-0.125 \cdot c\_n + -0.125 \cdot c\_p\right)\right)\right)}}} \]

13. Final simplification 99.6%

    \[\leadsto \frac{1}{e^{t \cdot \left(-0.5 \cdot c\_n + -0.5 \cdot c\_p\right) - s \cdot \left(-0.5 \cdot c\_n + \left(c\_p \cdot 0.5 + s \cdot \left(c\_n \cdot -0.125 + c\_p \cdot -0.125\right)\right)\right)}} \]
14. Add Preprocessing

Alternative 2: 98.4% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{e^{s \cdot \left(-0.5 \cdot \left(-c\_n\right) - \left(c\_p \cdot 0.5 + s \cdot \left(c\_n \cdot -0.125 + c\_p \cdot -0.125\right)\right)\right)}} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (/
  1.0
  (exp
   (*
    s
    (-
     (* -0.5 (- c_n))
     (+ (* c_p 0.5) (* s (+ (* c_n -0.125) (* c_p -0.125)))))))))

C

double code(double c_p, double c_n, double t, double s) {
	return 1.0 / exp((s * ((-0.5 * -c_n) - ((c_p * 0.5) + (s * ((c_n * -0.125) + (c_p * -0.125)))))));
}

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((s * (((-0.5d0) * -c_n) - ((c_p * 0.5d0) + (s * ((c_n * (-0.125d0)) + (c_p * (-0.125d0))))))))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return 1.0 / Math.exp((s * ((-0.5 * -c_n) - ((c_p * 0.5) + (s * ((c_n * -0.125) + (c_p * -0.125)))))));
}

Python

def code(c_p, c_n, t, s):
	return 1.0 / math.exp((s * ((-0.5 * -c_n) - ((c_p * 0.5) + (s * ((c_n * -0.125) + (c_p * -0.125)))))))

Julia

function code(c_p, c_n, t, s)
	return Float64(1.0 / exp(Float64(s * Float64(Float64(-0.5 * Float64(-c_n)) - Float64(Float64(c_p * 0.5) + Float64(s * Float64(Float64(c_n * -0.125) + Float64(c_p * -0.125))))))))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = 1.0 / exp((s * ((-0.5 * -c_n) - ((c_p * 0.5) + (s * ((c_n * -0.125) + (c_p * -0.125)))))));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[(1.0 / N[Exp[N[(s * N[(N[(-0.5 * (-c$95$n)), $MachinePrecision] - N[(N[(c$95$p * 0.5), $MachinePrecision] + N[(s * N[(N[(c$95$n * -0.125), $MachinePrecision] + N[(c$95$p * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.3%

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

   1. associate-/l/ 88.3%

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

3. Simplified 88.3%

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

   1. add-log-exp 87.9%

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

   2. inv-pow 87.9%

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

   3. pow-pow 87.9%

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

   4. add-sqr-sqrt 42.2%

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

   5. sqrt-unprod 87.9%

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

   6. sqr-neg 87.9%

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

   7. sqrt-unprod 45.7%

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

   8. add-sqr-sqrt 90.6%

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

6. Applied egg-rr 90.6%

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

8. Applied egg-rr 97.5%

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

   1. unpow-1 97.5%

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

10. Simplified 97.4%

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

11. Taylor expanded in s around 0 98.9%

    \[\leadsto \frac{1}{e^{\left(\mathsf{log1p}\left(e^{t}\right) \cdot \left(-c\_p\right) - \color{blue}{\left(-1 \cdot \left(c\_p \cdot \log 2\right) + \left(c\_n \cdot \log 0.5 + s \cdot \left(-0.5 \cdot c\_n + \left(0.5 \cdot c\_p + s \cdot \left(-0.125 \cdot c\_n + -0.125 \cdot c\_p\right)\right)\right)\right)\right)}\right) + c\_n \cdot \mathsf{log1p}\left(\frac{1}{-1 - e^{-t}}\right)}} \]

12. Taylor expanded in t around 0 98.4%

    \[\leadsto \frac{1}{e^{\color{blue}{-1 \cdot \left(s \cdot \left(-0.5 \cdot c\_n + \left(0.5 \cdot c\_p + s \cdot \left(-0.125 \cdot c\_n + -0.125 \cdot c\_p\right)\right)\right)\right)}}} \]

13. Final simplification 98.4%

    \[\leadsto \frac{1}{e^{s \cdot \left(-0.5 \cdot \left(-c\_n\right) - \left(c\_p \cdot 0.5 + s \cdot \left(c\_n \cdot -0.125 + c\_p \cdot -0.125\right)\right)\right)}} \]
14. Add Preprocessing

Alternative 3: 94.5% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ 1 + s \cdot \left(c\_p \cdot 0.5 + s \cdot \left(c\_p \cdot -0.125 + 0.125 \cdot {c\_p}^{2}\right)\right) \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (+
  1.0
  (* s (+ (* c_p 0.5) (* s (+ (* c_p -0.125) (* 0.125 (pow c_p 2.0))))))))

C

double code(double c_p, double c_n, double t, double s) {
	return 1.0 + (s * ((c_p * 0.5) + (s * ((c_p * -0.125) + (0.125 * pow(c_p, 2.0))))));
}

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 + (s * ((c_p * 0.5d0) + (s * ((c_p * (-0.125d0)) + (0.125d0 * (c_p ** 2.0d0))))))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return 1.0 + (s * ((c_p * 0.5) + (s * ((c_p * -0.125) + (0.125 * Math.pow(c_p, 2.0))))));
}

Python

def code(c_p, c_n, t, s):
	return 1.0 + (s * ((c_p * 0.5) + (s * ((c_p * -0.125) + (0.125 * math.pow(c_p, 2.0))))))

Julia

function code(c_p, c_n, t, s)
	return Float64(1.0 + Float64(s * Float64(Float64(c_p * 0.5) + Float64(s * Float64(Float64(c_p * -0.125) + Float64(0.125 * (c_p ^ 2.0)))))))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = 1.0 + (s * ((c_p * 0.5) + (s * ((c_p * -0.125) + (0.125 * (c_p ^ 2.0))))));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[(1.0 + N[(s * N[(N[(c$95$p * 0.5), $MachinePrecision] + N[(s * N[(N[(c$95$p * -0.125), $MachinePrecision] + N[(0.125 * N[Power[c$95$p, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.3%

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

   1. associate-/l/ 88.3%

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

3. Simplified 88.3%

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

5. Taylor expanded in c_n around 0 91.2%

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

6. Taylor expanded in s around 0 91.6%

   \[\leadsto \frac{{\left(\frac{1}{\color{blue}{2 + s \cdot \left(s \cdot \left(0.5 + -0.16666666666666666 \cdot s\right) - 1\right)}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p}} \]

7. Taylor expanded in t around 0 92.7%

   \[\leadsto \frac{{\left(\frac{1}{2 + s \cdot \left(s \cdot \left(0.5 + -0.16666666666666666 \cdot s\right) - 1\right)}\right)}^{c\_p}}{{\color{blue}{0.5}}^{c\_p}} \]

8. Taylor expanded in s around 0 94.4%

   \[\leadsto \color{blue}{1 + s \cdot \left(0.5 \cdot c\_p + s \cdot \left(-0.125 \cdot c\_p + 0.125 \cdot {c\_p}^{2}\right)\right)} \]

9. Final simplification 94.4%

   \[\leadsto 1 + s \cdot \left(c\_p \cdot 0.5 + s \cdot \left(c\_p \cdot -0.125 + 0.125 \cdot {c\_p}^{2}\right)\right) \]
10. Add Preprocessing

Alternative 4: 94.5% accurate, 75.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.3%

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

   1. associate-/l/ 88.3%

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

3. Simplified 88.3%

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

5. Taylor expanded in t around 0 90.2%

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

6. Taylor expanded in s around 0 94.4%

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

7. Final simplification 94.4%

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

Alternative 5: 94.5% accurate, 119.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.3%

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

   1. associate-/l/ 88.3%

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

3. Simplified 88.3%

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

5. Taylor expanded in t around 0 90.2%

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

6. Taylor expanded in s around 0 94.4%

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

7. Taylor expanded in c_n around 0 94.3%

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

   1. *-commutative 94.3%

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

   2. *-commutative 94.3%

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

   3. associate-*r* 94.3%

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

9. Simplified 94.3%

   \[\leadsto 1 + \color{blue}{s \cdot \left(c\_p \cdot 0.5\right)} \]
10. Add Preprocessing

Alternative 6: 94.5% accurate, 119.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.3%

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

   1. associate-/l/ 88.3%

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

3. Simplified 88.3%

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

5. Taylor expanded in t around 0 90.2%

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

6. Taylor expanded in s around 0 94.4%

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

7. Taylor expanded in c_n around inf 94.1%

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

   1. associate-*r* 94.1%

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

   2. *-commutative 94.1%

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

   3. *-commutative 94.1%

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

9. Simplified 94.1%

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

10. Final simplification 94.1%

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

Alternative 7: 94.5% 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 88.3%

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

   1. associate-/l/ 88.3%

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

3. Simplified 88.3%

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

5. Taylor expanded in c_n around 0 91.2%

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

6. Taylor expanded in c_p around 0 94.1%

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

Developer Target 1: 96.2% 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 2024154 
(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))))