Harley's example

Percentage Accurate: 91.0% → 99.6%

Time: 50.2s

Alternatives: 7

Speedup: 896.0×

• could not determine a ground truth

  1. c_p = 8.674421414310918e-42
  2. c_n = 2.8257158229687884e-272
  3. t = 7.448034528831689e+79
  4. s = 4.172938683981005e+124

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: 91.0% 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: 99.6% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ e^{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(c\_p + c\_n\right) \cdot \mathsf{fma}\left(t \cdot t, -0.005208333333333333, 0.125\right), 0.5 \cdot \left(c\_n - c\_p\right)\right), s \cdot \mathsf{fma}\left(s, \left(c\_p + c\_n\right) \cdot -0.125, \left(c\_n - c\_p\right) \cdot -0.5\right)\right)} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (exp
  (fma
   t
   (fma
    t
    (* (+ c_p c_n) (fma (* t t) -0.005208333333333333 0.125))
    (* 0.5 (- c_n c_p)))
   (* s (fma s (* (+ c_p c_n) -0.125) (* (- c_n c_p) -0.5))))))

C

double code(double c_p, double c_n, double t, double s) {
	return exp(fma(t, fma(t, ((c_p + c_n) * fma((t * t), -0.005208333333333333, 0.125)), (0.5 * (c_n - c_p))), (s * fma(s, ((c_p + c_n) * -0.125), ((c_n - c_p) * -0.5)))));
}

Julia

function code(c_p, c_n, t, s)
	return exp(fma(t, fma(t, Float64(Float64(c_p + c_n) * fma(Float64(t * t), -0.005208333333333333, 0.125)), Float64(0.5 * Float64(c_n - c_p))), Float64(s * fma(s, Float64(Float64(c_p + c_n) * -0.125), Float64(Float64(c_n - c_p) * -0.5)))))
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[Exp[N[(t * N[(t * N[(N[(c$95$p + c$95$n), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * -0.005208333333333333 + 0.125), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c$95$n - c$95$p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(s * N[(s * N[(N[(c$95$p + c$95$n), $MachinePrecision] * -0.125), $MachinePrecision] + N[(N[(c$95$n - c$95$p), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 91.0%

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

4. Applied egg-rr 97.6%

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

5. Taylor expanded in t around 0

   \[\leadsto e^{\color{blue}{\left(-1 \cdot \left(c\_p \cdot \log \left(1 + e^{\mathsf{neg}\left(s\right)}\right)\right) + \left(c\_n \cdot \log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) + t \cdot \left(t \cdot \left(-1 \cdot \left({t}^{2} \cdot \left(\frac{1}{192} \cdot c\_n + \frac{1}{192} \cdot c\_p\right)\right) - \left(\frac{-1}{8} \cdot c\_n + \frac{-1}{8} \cdot c\_p\right)\right) - \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) - \left(-1 \cdot \left(c\_p \cdot \log 2\right) + c\_n \cdot \log \frac{1}{2}\right)}} \]

7. Simplified 99.9%

   \[\leadsto e^{\color{blue}{\left(\mathsf{fma}\left(c\_n, \mathsf{log1p}\left(\frac{-1}{1 + e^{-s}}\right), t \cdot \mathsf{fma}\left(t, 0.125 \cdot \left(c\_n + c\_p\right) - \left(0.005208333333333333 \cdot \left(c\_n + c\_p\right)\right) \cdot \left(t \cdot t\right), -\mathsf{fma}\left(c\_n, -0.5, c\_p \cdot 0.5\right)\right)\right) - c\_p \cdot \mathsf{log1p}\left(e^{-s}\right)\right) - \mathsf{fma}\left(\log 2, -c\_p, c\_n \cdot \log 0.5\right)}} \]

8. Taylor expanded in s around 0

   \[\leadsto e^{\color{blue}{\left(s \cdot \left(\left(\frac{-1}{2} \cdot c\_n + s \cdot \left(\frac{-1}{8} \cdot c\_n - \frac{1}{8} \cdot c\_p\right)\right) - \frac{-1}{2} \cdot c\_p\right) + t \cdot \left(t \cdot \left(\frac{1}{8} \cdot \left(c\_n + c\_p\right) - \frac{1}{192} \cdot \left({t}^{2} \cdot \left(c\_n + c\_p\right)\right)\right) - \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right) - \left(-1 \cdot \left(c\_p \cdot \log 2\right) + c\_p \cdot \log 2\right)}} \]

10. Simplified 100.0%

    \[\leadsto e^{\color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(c\_p + c\_n\right) \cdot \mathsf{fma}\left(t \cdot t, -0.005208333333333333, 0.125\right), 0.5 \cdot \left(c\_n - c\_p\right)\right), s \cdot \mathsf{fma}\left(s, -0.125 \cdot \left(c\_p + c\_n\right), -0.5 \cdot \left(c\_n - c\_p\right)\right)\right)}} \]

11. Final simplification 100.0%

    \[\leadsto e^{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(c\_p + c\_n\right) \cdot \mathsf{fma}\left(t \cdot t, -0.005208333333333333, 0.125\right), 0.5 \cdot \left(c\_n - c\_p\right)\right), s \cdot \mathsf{fma}\left(s, \left(c\_p + c\_n\right) \cdot -0.125, \left(c\_n - c\_p\right) \cdot -0.5\right)\right)} \]
12. Add Preprocessing

Alternative 2: 98.8% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ e^{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(t \cdot t, 0.005208333333333333, -0.125\right), -0.5\right), s \cdot \mathsf{fma}\left(s, 0.125, 0.5\right)\right) \cdot \left(-c\_n\right)} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (exp
  (*
   (fma
    t
    (fma t (fma (* t t) 0.005208333333333333 -0.125) -0.5)
    (* s (fma s 0.125 0.5)))
   (- c_n))))

C

double code(double c_p, double c_n, double t, double s) {
	return exp((fma(t, fma(t, fma((t * t), 0.005208333333333333, -0.125), -0.5), (s * fma(s, 0.125, 0.5))) * -c_n));
}

Julia

function code(c_p, c_n, t, s)
	return exp(Float64(fma(t, fma(t, fma(Float64(t * t), 0.005208333333333333, -0.125), -0.5), Float64(s * fma(s, 0.125, 0.5))) * Float64(-c_n)))
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[Exp[N[(N[(t * N[(t * N[(N[(t * t), $MachinePrecision] * 0.005208333333333333 + -0.125), $MachinePrecision] + -0.5), $MachinePrecision] + N[(s * N[(s * 0.125 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-c$95$n)), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 91.0%

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

4. Applied egg-rr 97.6%

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

5. Taylor expanded in t around 0

   \[\leadsto e^{\color{blue}{\left(-1 \cdot \left(c\_p \cdot \log \left(1 + e^{\mathsf{neg}\left(s\right)}\right)\right) + \left(c\_n \cdot \log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) + t \cdot \left(t \cdot \left(-1 \cdot \left({t}^{2} \cdot \left(\frac{1}{192} \cdot c\_n + \frac{1}{192} \cdot c\_p\right)\right) - \left(\frac{-1}{8} \cdot c\_n + \frac{-1}{8} \cdot c\_p\right)\right) - \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) - \left(-1 \cdot \left(c\_p \cdot \log 2\right) + c\_n \cdot \log \frac{1}{2}\right)}} \]

7. Simplified 99.9%

   \[\leadsto e^{\color{blue}{\left(\mathsf{fma}\left(c\_n, \mathsf{log1p}\left(\frac{-1}{1 + e^{-s}}\right), t \cdot \mathsf{fma}\left(t, 0.125 \cdot \left(c\_n + c\_p\right) - \left(0.005208333333333333 \cdot \left(c\_n + c\_p\right)\right) \cdot \left(t \cdot t\right), -\mathsf{fma}\left(c\_n, -0.5, c\_p \cdot 0.5\right)\right)\right) - c\_p \cdot \mathsf{log1p}\left(e^{-s}\right)\right) - \mathsf{fma}\left(\log 2, -c\_p, c\_n \cdot \log 0.5\right)}} \]

8. Taylor expanded in s around 0

   \[\leadsto e^{\color{blue}{\left(s \cdot \left(\left(\frac{-1}{2} \cdot c\_n + s \cdot \left(\frac{-1}{8} \cdot c\_n - \frac{1}{8} \cdot c\_p\right)\right) - \frac{-1}{2} \cdot c\_p\right) + t \cdot \left(t \cdot \left(\frac{1}{8} \cdot \left(c\_n + c\_p\right) - \frac{1}{192} \cdot \left({t}^{2} \cdot \left(c\_n + c\_p\right)\right)\right) - \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right) - \left(-1 \cdot \left(c\_p \cdot \log 2\right) + c\_p \cdot \log 2\right)}} \]

10. Simplified 100.0%

    \[\leadsto e^{\color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(c\_p + c\_n\right) \cdot \mathsf{fma}\left(t \cdot t, -0.005208333333333333, 0.125\right), 0.5 \cdot \left(c\_n - c\_p\right)\right), s \cdot \mathsf{fma}\left(s, -0.125 \cdot \left(c\_p + c\_n\right), -0.5 \cdot \left(c\_n - c\_p\right)\right)\right)}} \]

11. Taylor expanded in c_n around -inf

    \[\leadsto e^{\color{blue}{-1 \cdot \left(c\_n \cdot \left(s \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot s\right) + t \cdot \left(-1 \cdot \left(t \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {t}^{2}\right)\right) - \frac{1}{2}\right)\right)\right)}} \]
12. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto e^{\color{blue}{\mathsf{neg}\left(c\_n \cdot \left(s \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot s\right) + t \cdot \left(-1 \cdot \left(t \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {t}^{2}\right)\right) - \frac{1}{2}\right)\right)\right)}} \]

    2. *-commutative N/A

       \[\leadsto e^{\mathsf{neg}\left(\color{blue}{\left(s \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot s\right) + t \cdot \left(-1 \cdot \left(t \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {t}^{2}\right)\right) - \frac{1}{2}\right)\right) \cdot c\_n}\right)} \]

    3. distribute-rgt-neg-in N/A

       \[\leadsto e^{\color{blue}{\left(s \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot s\right) + t \cdot \left(-1 \cdot \left(t \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {t}^{2}\right)\right) - \frac{1}{2}\right)\right) \cdot \left(\mathsf{neg}\left(c\_n\right)\right)}} \]

    4. mul-1-neg N/A

       \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot s\right) + t \cdot \left(-1 \cdot \left(t \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {t}^{2}\right)\right) - \frac{1}{2}\right)\right) \cdot \color{blue}{\left(-1 \cdot c\_n\right)}} \]

    5. lower-*.f64 N/A

       \[\leadsto e^{\color{blue}{\left(s \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot s\right) + t \cdot \left(-1 \cdot \left(t \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {t}^{2}\right)\right) - \frac{1}{2}\right)\right) \cdot \left(-1 \cdot c\_n\right)}} \]

13. Simplified 99.5%

    \[\leadsto e^{\color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(t \cdot t, 0.005208333333333333, -0.125\right), -0.5\right), s \cdot \mathsf{fma}\left(s, 0.125, 0.5\right)\right) \cdot \left(-c\_n\right)}} \]
14. Add Preprocessing

Alternative 3: 98.8% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ e^{c\_n \cdot \mathsf{fma}\left(s, \mathsf{fma}\left(s, -0.125, -0.5\right), t \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, t \cdot -0.005208333333333333, 0.125\right), 0.5\right)\right)} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (exp
  (*
   c_n
   (fma
    s
    (fma s -0.125 -0.5)
    (* t (fma t (fma t (* t -0.005208333333333333) 0.125) 0.5))))))

C

double code(double c_p, double c_n, double t, double s) {
	return exp((c_n * fma(s, fma(s, -0.125, -0.5), (t * fma(t, fma(t, (t * -0.005208333333333333), 0.125), 0.5)))));
}

Julia

function code(c_p, c_n, t, s)
	return exp(Float64(c_n * fma(s, fma(s, -0.125, -0.5), Float64(t * fma(t, fma(t, Float64(t * -0.005208333333333333), 0.125), 0.5)))))
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[Exp[N[(c$95$n * N[(s * N[(s * -0.125 + -0.5), $MachinePrecision] + N[(t * N[(t * N[(t * N[(t * -0.005208333333333333), $MachinePrecision] + 0.125), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 91.0%

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

4. Applied egg-rr 97.6%

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

5. Taylor expanded in t around 0

   \[\leadsto e^{\color{blue}{\left(-1 \cdot \left(c\_p \cdot \log \left(1 + e^{\mathsf{neg}\left(s\right)}\right)\right) + \left(c\_n \cdot \log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) + t \cdot \left(t \cdot \left(-1 \cdot \left({t}^{2} \cdot \left(\frac{1}{192} \cdot c\_n + \frac{1}{192} \cdot c\_p\right)\right) - \left(\frac{-1}{8} \cdot c\_n + \frac{-1}{8} \cdot c\_p\right)\right) - \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) - \left(-1 \cdot \left(c\_p \cdot \log 2\right) + c\_n \cdot \log \frac{1}{2}\right)}} \]

7. Simplified 99.9%

   \[\leadsto e^{\color{blue}{\left(\mathsf{fma}\left(c\_n, \mathsf{log1p}\left(\frac{-1}{1 + e^{-s}}\right), t \cdot \mathsf{fma}\left(t, 0.125 \cdot \left(c\_n + c\_p\right) - \left(0.005208333333333333 \cdot \left(c\_n + c\_p\right)\right) \cdot \left(t \cdot t\right), -\mathsf{fma}\left(c\_n, -0.5, c\_p \cdot 0.5\right)\right)\right) - c\_p \cdot \mathsf{log1p}\left(e^{-s}\right)\right) - \mathsf{fma}\left(\log 2, -c\_p, c\_n \cdot \log 0.5\right)}} \]

8. Taylor expanded in s around 0

   \[\leadsto e^{\color{blue}{\left(s \cdot \left(\left(\frac{-1}{2} \cdot c\_n + s \cdot \left(\frac{-1}{8} \cdot c\_n - \frac{1}{8} \cdot c\_p\right)\right) - \frac{-1}{2} \cdot c\_p\right) + t \cdot \left(t \cdot \left(\frac{1}{8} \cdot \left(c\_n + c\_p\right) - \frac{1}{192} \cdot \left({t}^{2} \cdot \left(c\_n + c\_p\right)\right)\right) - \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right) - \left(-1 \cdot \left(c\_p \cdot \log 2\right) + c\_p \cdot \log 2\right)}} \]

10. Simplified 100.0%

    \[\leadsto e^{\color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(c\_p + c\_n\right) \cdot \mathsf{fma}\left(t \cdot t, -0.005208333333333333, 0.125\right), 0.5 \cdot \left(c\_n - c\_p\right)\right), s \cdot \mathsf{fma}\left(s, -0.125 \cdot \left(c\_p + c\_n\right), -0.5 \cdot \left(c\_n - c\_p\right)\right)\right)}} \]

11. Taylor expanded in c_p around 0

    \[\leadsto e^{\color{blue}{s \cdot \left(\frac{-1}{2} \cdot c\_n + \frac{-1}{8} \cdot \left(c\_n \cdot s\right)\right) + t \cdot \left(\frac{1}{2} \cdot c\_n + c\_n \cdot \left(t \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {t}^{2}\right)\right)\right)}} \]
12. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto e^{s \cdot \color{blue}{\left(\frac{-1}{8} \cdot \left(c\_n \cdot s\right) + \frac{-1}{2} \cdot c\_n\right)} + t \cdot \left(\frac{1}{2} \cdot c\_n + c\_n \cdot \left(t \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {t}^{2}\right)\right)\right)} \]

    2. distribute-rgt-in N/A

       \[\leadsto e^{\color{blue}{\left(\left(\frac{-1}{8} \cdot \left(c\_n \cdot s\right)\right) \cdot s + \left(\frac{-1}{2} \cdot c\_n\right) \cdot s\right)} + t \cdot \left(\frac{1}{2} \cdot c\_n + c\_n \cdot \left(t \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {t}^{2}\right)\right)\right)} \]

    3. *-commutative N/A

       \[\leadsto e^{\left(\color{blue}{\left(\left(c\_n \cdot s\right) \cdot \frac{-1}{8}\right)} \cdot s + \left(\frac{-1}{2} \cdot c\_n\right) \cdot s\right) + t \cdot \left(\frac{1}{2} \cdot c\_n + c\_n \cdot \left(t \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {t}^{2}\right)\right)\right)} \]

    4. associate-*l* N/A

       \[\leadsto e^{\left(\color{blue}{\left(c\_n \cdot s\right) \cdot \left(\frac{-1}{8} \cdot s\right)} + \left(\frac{-1}{2} \cdot c\_n\right) \cdot s\right) + t \cdot \left(\frac{1}{2} \cdot c\_n + c\_n \cdot \left(t \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {t}^{2}\right)\right)\right)} \]

    5. associate-*r* N/A

       \[\leadsto e^{\left(\left(c\_n \cdot s\right) \cdot \left(\frac{-1}{8} \cdot s\right) + \color{blue}{\frac{-1}{2} \cdot \left(c\_n \cdot s\right)}\right) + t \cdot \left(\frac{1}{2} \cdot c\_n + c\_n \cdot \left(t \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {t}^{2}\right)\right)\right)} \]

    6. *-commutative N/A

       \[\leadsto e^{\left(\left(c\_n \cdot s\right) \cdot \left(\frac{-1}{8} \cdot s\right) + \color{blue}{\left(c\_n \cdot s\right) \cdot \frac{-1}{2}}\right) + t \cdot \left(\frac{1}{2} \cdot c\_n + c\_n \cdot \left(t \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {t}^{2}\right)\right)\right)} \]

    7. distribute-lft-in N/A

       \[\leadsto e^{\color{blue}{\left(c\_n \cdot s\right) \cdot \left(\frac{-1}{8} \cdot s + \frac{-1}{2}\right)} + t \cdot \left(\frac{1}{2} \cdot c\_n + c\_n \cdot \left(t \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {t}^{2}\right)\right)\right)} \]

    8. metadata-eval N/A

       \[\leadsto e^{\left(c\_n \cdot s\right) \cdot \left(\frac{-1}{8} \cdot s + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) + t \cdot \left(\frac{1}{2} \cdot c\_n + c\_n \cdot \left(t \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {t}^{2}\right)\right)\right)} \]

    9. sub-neg N/A

       \[\leadsto e^{\left(c\_n \cdot s\right) \cdot \color{blue}{\left(\frac{-1}{8} \cdot s - \frac{1}{2}\right)} + t \cdot \left(\frac{1}{2} \cdot c\_n + c\_n \cdot \left(t \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {t}^{2}\right)\right)\right)} \]

    10. associate-*r* N/A

        \[\leadsto e^{\color{blue}{c\_n \cdot \left(s \cdot \left(\frac{-1}{8} \cdot s - \frac{1}{2}\right)\right)} + t \cdot \left(\frac{1}{2} \cdot c\_n + c\_n \cdot \left(t \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {t}^{2}\right)\right)\right)} \]

    11. *-commutative N/A

        \[\leadsto e^{\color{blue}{\left(s \cdot \left(\frac{-1}{8} \cdot s - \frac{1}{2}\right)\right) \cdot c\_n} + t \cdot \left(\frac{1}{2} \cdot c\_n + c\_n \cdot \left(t \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {t}^{2}\right)\right)\right)} \]

    12. *-commutative N/A

        \[\leadsto e^{\left(s \cdot \left(\frac{-1}{8} \cdot s - \frac{1}{2}\right)\right) \cdot c\_n + t \cdot \left(\frac{1}{2} \cdot c\_n + \color{blue}{\left(t \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {t}^{2}\right)\right) \cdot c\_n}\right)} \]

    13. distribute-rgt-in N/A

        \[\leadsto e^{\left(s \cdot \left(\frac{-1}{8} \cdot s - \frac{1}{2}\right)\right) \cdot c\_n + t \cdot \color{blue}{\left(c\_n \cdot \left(\frac{1}{2} + t \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {t}^{2}\right)\right)\right)}} \]

    14. *-commutative N/A

        \[\leadsto e^{\left(s \cdot \left(\frac{-1}{8} \cdot s - \frac{1}{2}\right)\right) \cdot c\_n + t \cdot \color{blue}{\left(\left(\frac{1}{2} + t \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {t}^{2}\right)\right) \cdot c\_n\right)}} \]

    15. associate-*r* N/A

        \[\leadsto e^{\left(s \cdot \left(\frac{-1}{8} \cdot s - \frac{1}{2}\right)\right) \cdot c\_n + \color{blue}{\left(t \cdot \left(\frac{1}{2} + t \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {t}^{2}\right)\right)\right) \cdot c\_n}} \]

    16. distribute-rgt-in N/A

        \[\leadsto e^{\color{blue}{c\_n \cdot \left(s \cdot \left(\frac{-1}{8} \cdot s - \frac{1}{2}\right) + t \cdot \left(\frac{1}{2} + t \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {t}^{2}\right)\right)\right)}} \]

13. Simplified 99.5%

    \[\leadsto e^{\color{blue}{c\_n \cdot \mathsf{fma}\left(s, \mathsf{fma}\left(s, -0.125, -0.5\right), t \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, t \cdot -0.005208333333333333, 0.125\right), 0.5\right)\right)}} \]
14. Add Preprocessing

Alternative 4: 98.3% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-t \leq 4 \cdot 10^{-163}:\\ \;\;\;\;e^{s \cdot \left(\left(c\_n - c\_p\right) \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{t \cdot \left(-0.5 \cdot \left(c\_p - c\_n\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- t) 4e-163)
   (exp (* s (* (- c_n c_p) -0.5)))
   (exp (* t (* -0.5 (- c_p c_n))))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-t <= 4e-163) {
		tmp = exp((s * ((c_n - c_p) * -0.5)));
	} else {
		tmp = exp((t * (-0.5 * (c_p - c_n))));
	}
	return tmp;
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    real(8) :: tmp
    if (-t <= 4d-163) then
        tmp = exp((s * ((c_n - c_p) * (-0.5d0))))
    else
        tmp = exp((t * ((-0.5d0) * (c_p - c_n))))
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-t <= 4e-163) {
		tmp = Math.exp((s * ((c_n - c_p) * -0.5)));
	} else {
		tmp = Math.exp((t * (-0.5 * (c_p - c_n))));
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if -t <= 4e-163:
		tmp = math.exp((s * ((c_n - c_p) * -0.5)))
	else:
		tmp = math.exp((t * (-0.5 * (c_p - c_n))))
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (Float64(-t) <= 4e-163)
		tmp = exp(Float64(s * Float64(Float64(c_n - c_p) * -0.5)));
	else
		tmp = exp(Float64(t * Float64(-0.5 * Float64(c_p - c_n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (-t <= 4e-163)
		tmp = exp((s * ((c_n - c_p) * -0.5)));
	else
		tmp = exp((t * (-0.5 * (c_p - c_n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-t), 4e-163], N[Exp[N[(s * N[(N[(c$95$n - c$95$p), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(t * N[(-0.5 * N[(c$95$p - c$95$n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 t) < 3.99999999999999969e-163

   1. Initial program 94.1%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in t around 0

      \[\leadsto e^{\color{blue}{\left(-1 \cdot \left(c\_p \cdot \log \left(1 + e^{\mathsf{neg}\left(s\right)}\right)\right) + \left(c\_n \cdot \log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) + t \cdot \left(t \cdot \left(-1 \cdot \left({t}^{2} \cdot \left(\frac{1}{192} \cdot c\_n + \frac{1}{192} \cdot c\_p\right)\right) - \left(\frac{-1}{8} \cdot c\_n + \frac{-1}{8} \cdot c\_p\right)\right) - \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) - \left(-1 \cdot \left(c\_p \cdot \log 2\right) + c\_n \cdot \log \frac{1}{2}\right)}} \]

   7. Simplified 100.0%

      \[\leadsto e^{\color{blue}{\left(\mathsf{fma}\left(c\_n, \mathsf{log1p}\left(\frac{-1}{1 + e^{-s}}\right), t \cdot \mathsf{fma}\left(t, 0.125 \cdot \left(c\_n + c\_p\right) - \left(0.005208333333333333 \cdot \left(c\_n + c\_p\right)\right) \cdot \left(t \cdot t\right), -\mathsf{fma}\left(c\_n, -0.5, c\_p \cdot 0.5\right)\right)\right) - c\_p \cdot \mathsf{log1p}\left(e^{-s}\right)\right) - \mathsf{fma}\left(\log 2, -c\_p, c\_n \cdot \log 0.5\right)}} \]

   8. Taylor expanded in s around 0

      \[\leadsto e^{\color{blue}{\left(s \cdot \left(\frac{-1}{2} \cdot c\_n - \frac{-1}{2} \cdot c\_p\right) + t \cdot \left(t \cdot \left(\frac{1}{8} \cdot \left(c\_n + c\_p\right) - \frac{1}{192} \cdot \left({t}^{2} \cdot \left(c\_n + c\_p\right)\right)\right) - \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right) - \left(-1 \cdot \left(c\_p \cdot \log 2\right) + c\_p \cdot \log 2\right)}} \]

   10. Simplified 99.4%

       \[\leadsto e^{\color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(c\_p + c\_n\right) \cdot \mathsf{fma}\left(t \cdot t, -0.005208333333333333, 0.125\right), 0.5 \cdot \left(c\_n - c\_p\right)\right), s \cdot \left(-0.5 \cdot \left(c\_n - c\_p\right)\right)\right)}} \]

   11. Taylor expanded in t around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower--.f64 99.4

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

   13. Simplified 99.4%

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

   if 3.99999999999999969e-163 < (neg.f64 t)

   1. Initial program 82.6%

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

   4. Applied egg-rr 91.0%

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

   5. Taylor expanded in t around 0

      \[\leadsto e^{\color{blue}{\left(-1 \cdot \left(c\_p \cdot \log \left(1 + e^{\mathsf{neg}\left(s\right)}\right)\right) + \left(c\_n \cdot \log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) + t \cdot \left(t \cdot \left(-1 \cdot \left({t}^{2} \cdot \left(\frac{1}{192} \cdot c\_n + \frac{1}{192} \cdot c\_p\right)\right) - \left(\frac{-1}{8} \cdot c\_n + \frac{-1}{8} \cdot c\_p\right)\right) - \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) - \left(-1 \cdot \left(c\_p \cdot \log 2\right) + c\_n \cdot \log \frac{1}{2}\right)}} \]

   7. Simplified 99.6%

      \[\leadsto e^{\color{blue}{\left(\mathsf{fma}\left(c\_n, \mathsf{log1p}\left(\frac{-1}{1 + e^{-s}}\right), t \cdot \mathsf{fma}\left(t, 0.125 \cdot \left(c\_n + c\_p\right) - \left(0.005208333333333333 \cdot \left(c\_n + c\_p\right)\right) \cdot \left(t \cdot t\right), -\mathsf{fma}\left(c\_n, -0.5, c\_p \cdot 0.5\right)\right)\right) - c\_p \cdot \mathsf{log1p}\left(e^{-s}\right)\right) - \mathsf{fma}\left(\log 2, -c\_p, c\_n \cdot \log 0.5\right)}} \]

   8. Taylor expanded in s around 0

      \[\leadsto e^{\color{blue}{t \cdot \left(t \cdot \left(\frac{1}{8} \cdot \left(c\_n + c\_p\right) - \frac{1}{192} \cdot \left({t}^{2} \cdot \left(c\_n + c\_p\right)\right)\right) - \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right) - \left(-1 \cdot \left(c\_p \cdot \log 2\right) + c\_p \cdot \log 2\right)}} \]
   9. Step-by-step derivation

      1. distribute-lft1-in N/A

         \[\leadsto e^{t \cdot \left(t \cdot \left(\frac{1}{8} \cdot \left(c\_n + c\_p\right) - \frac{1}{192} \cdot \left({t}^{2} \cdot \left(c\_n + c\_p\right)\right)\right) - \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right) - \color{blue}{\left(-1 + 1\right) \cdot \left(c\_p \cdot \log 2\right)}} \]

      2. metadata-eval N/A

         \[\leadsto e^{t \cdot \left(t \cdot \left(\frac{1}{8} \cdot \left(c\_n + c\_p\right) - \frac{1}{192} \cdot \left({t}^{2} \cdot \left(c\_n + c\_p\right)\right)\right) - \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right) - \color{blue}{0} \cdot \left(c\_p \cdot \log 2\right)} \]

      3. mul0-lft N/A

         \[\leadsto e^{t \cdot \left(t \cdot \left(\frac{1}{8} \cdot \left(c\_n + c\_p\right) - \frac{1}{192} \cdot \left({t}^{2} \cdot \left(c\_n + c\_p\right)\right)\right) - \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right) - \color{blue}{0}} \]

      4. --rgt-identity N/A

         \[\leadsto e^{\color{blue}{t \cdot \left(t \cdot \left(\frac{1}{8} \cdot \left(c\_n + c\_p\right) - \frac{1}{192} \cdot \left({t}^{2} \cdot \left(c\_n + c\_p\right)\right)\right) - \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)}} \]

      5. sub-neg N/A

         \[\leadsto e^{t \cdot \color{blue}{\left(t \cdot \left(\frac{1}{8} \cdot \left(c\_n + c\_p\right) - \frac{1}{192} \cdot \left({t}^{2} \cdot \left(c\_n + c\_p\right)\right)\right) + \left(\mathsf{neg}\left(\left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right)}} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto e^{t \cdot \left(t \cdot \color{blue}{\left(\frac{1}{8} \cdot \left(c\_n + c\_p\right) + \left(\mathsf{neg}\left(\frac{1}{192}\right)\right) \cdot \left({t}^{2} \cdot \left(c\_n + c\_p\right)\right)\right)} + \left(\mathsf{neg}\left(\left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right)} \]

      7. metadata-eval N/A

         \[\leadsto e^{t \cdot \left(t \cdot \left(\frac{1}{8} \cdot \left(c\_n + c\_p\right) + \color{blue}{\frac{-1}{192}} \cdot \left({t}^{2} \cdot \left(c\_n + c\_p\right)\right)\right) + \left(\mathsf{neg}\left(\left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto e^{t \cdot \left(t \cdot \color{blue}{\left(\frac{-1}{192} \cdot \left({t}^{2} \cdot \left(c\_n + c\_p\right)\right) + \frac{1}{8} \cdot \left(c\_n + c\_p\right)\right)} + \left(\mathsf{neg}\left(\left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right)} \]

      9. mul-1-neg N/A

         \[\leadsto e^{t \cdot \left(t \cdot \left(\frac{-1}{192} \cdot \left({t}^{2} \cdot \left(c\_n + c\_p\right)\right) + \frac{1}{8} \cdot \left(c\_n + c\_p\right)\right) + \color{blue}{-1 \cdot \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)}\right)} \]

      10. +-commutative N/A

          \[\leadsto e^{t \cdot \color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right) + t \cdot \left(\frac{-1}{192} \cdot \left({t}^{2} \cdot \left(c\_n + c\_p\right)\right) + \frac{1}{8} \cdot \left(c\_n + c\_p\right)\right)\right)}} \]

   10. Simplified 99.0%

       \[\leadsto e^{\color{blue}{t \cdot \mathsf{fma}\left(t, \left(c\_p + c\_n\right) \cdot \mathsf{fma}\left(t \cdot t, -0.005208333333333333, 0.125\right), 0.5 \cdot \left(c\_n - c\_p\right)\right)}} \]

   11. Taylor expanded in t around 0

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

       1. sub-neg N/A

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

       2. mul-1-neg N/A

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

       3. +-commutative N/A

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

       4. distribute-lft-in N/A

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

       5. associate-*r* N/A

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

       6. metadata-eval N/A

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

       7. metadata-eval N/A

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

       8. associate-*r* N/A

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

       9. distribute-lft-in N/A

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

       10. lower-*.f64 N/A

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

       11. mul-1-neg N/A

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

       12. unsub-neg N/A

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

       13. lower--.f64 99.1

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

   13. Simplified 99.1%

       \[\leadsto e^{t \cdot \color{blue}{\left(-0.5 \cdot \left(c\_p - c\_n\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-t \leq 4 \cdot 10^{-163}:\\ \;\;\;\;e^{s \cdot \left(\left(c\_n - c\_p\right) \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{t \cdot \left(-0.5 \cdot \left(c\_p - c\_n\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

4. Applied egg-rr 97.6%

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

5. Taylor expanded in t around 0

   \[\leadsto e^{\color{blue}{\left(-1 \cdot \left(c\_p \cdot \log \left(1 + e^{\mathsf{neg}\left(s\right)}\right)\right) + \left(c\_n \cdot \log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) + t \cdot \left(t \cdot \left(-1 \cdot \left({t}^{2} \cdot \left(\frac{1}{192} \cdot c\_n + \frac{1}{192} \cdot c\_p\right)\right) - \left(\frac{-1}{8} \cdot c\_n + \frac{-1}{8} \cdot c\_p\right)\right) - \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) - \left(-1 \cdot \left(c\_p \cdot \log 2\right) + c\_n \cdot \log \frac{1}{2}\right)}} \]

7. Simplified 99.9%

   \[\leadsto e^{\color{blue}{\left(\mathsf{fma}\left(c\_n, \mathsf{log1p}\left(\frac{-1}{1 + e^{-s}}\right), t \cdot \mathsf{fma}\left(t, 0.125 \cdot \left(c\_n + c\_p\right) - \left(0.005208333333333333 \cdot \left(c\_n + c\_p\right)\right) \cdot \left(t \cdot t\right), -\mathsf{fma}\left(c\_n, -0.5, c\_p \cdot 0.5\right)\right)\right) - c\_p \cdot \mathsf{log1p}\left(e^{-s}\right)\right) - \mathsf{fma}\left(\log 2, -c\_p, c\_n \cdot \log 0.5\right)}} \]

8. Taylor expanded in s around 0

   \[\leadsto e^{\color{blue}{\left(s \cdot \left(\frac{-1}{2} \cdot c\_n - \frac{-1}{2} \cdot c\_p\right) + t \cdot \left(t \cdot \left(\frac{1}{8} \cdot \left(c\_n + c\_p\right) - \frac{1}{192} \cdot \left({t}^{2} \cdot \left(c\_n + c\_p\right)\right)\right) - \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right) - \left(-1 \cdot \left(c\_p \cdot \log 2\right) + c\_p \cdot \log 2\right)}} \]

10. Simplified 99.5%

    \[\leadsto e^{\color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(c\_p + c\_n\right) \cdot \mathsf{fma}\left(t \cdot t, -0.005208333333333333, 0.125\right), 0.5 \cdot \left(c\_n - c\_p\right)\right), s \cdot \left(-0.5 \cdot \left(c\_n - c\_p\right)\right)\right)}} \]

11. Taylor expanded in t around 0

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

    1. +-commutative N/A

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

    2. associate-*r* N/A

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

    3. associate-*r* N/A

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

    4. distribute-rgt-out N/A

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

    5. lower-*.f64 N/A

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

    6. lower--.f64 N/A

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

    7. lower-fma.f64 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f64 99.3

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

13. Simplified 99.3%

    \[\leadsto e^{\color{blue}{\left(c\_n - c\_p\right) \cdot \mathsf{fma}\left(0.5, t, s \cdot -0.5\right)}} \]
14. Add Preprocessing

Alternative 6: 98.4% accurate, 7.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

4. Applied egg-rr 97.6%

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

5. Taylor expanded in t around 0

   \[\leadsto e^{\color{blue}{\left(-1 \cdot \left(c\_p \cdot \log \left(1 + e^{\mathsf{neg}\left(s\right)}\right)\right) + \left(c\_n \cdot \log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) + t \cdot \left(t \cdot \left(-1 \cdot \left({t}^{2} \cdot \left(\frac{1}{192} \cdot c\_n + \frac{1}{192} \cdot c\_p\right)\right) - \left(\frac{-1}{8} \cdot c\_n + \frac{-1}{8} \cdot c\_p\right)\right) - \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) - \left(-1 \cdot \left(c\_p \cdot \log 2\right) + c\_n \cdot \log \frac{1}{2}\right)}} \]

7. Simplified 99.9%

   \[\leadsto e^{\color{blue}{\left(\mathsf{fma}\left(c\_n, \mathsf{log1p}\left(\frac{-1}{1 + e^{-s}}\right), t \cdot \mathsf{fma}\left(t, 0.125 \cdot \left(c\_n + c\_p\right) - \left(0.005208333333333333 \cdot \left(c\_n + c\_p\right)\right) \cdot \left(t \cdot t\right), -\mathsf{fma}\left(c\_n, -0.5, c\_p \cdot 0.5\right)\right)\right) - c\_p \cdot \mathsf{log1p}\left(e^{-s}\right)\right) - \mathsf{fma}\left(\log 2, -c\_p, c\_n \cdot \log 0.5\right)}} \]

8. Taylor expanded in s around 0

   \[\leadsto e^{\color{blue}{\left(s \cdot \left(\frac{-1}{2} \cdot c\_n - \frac{-1}{2} \cdot c\_p\right) + t \cdot \left(t \cdot \left(\frac{1}{8} \cdot \left(c\_n + c\_p\right) - \frac{1}{192} \cdot \left({t}^{2} \cdot \left(c\_n + c\_p\right)\right)\right) - \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right) - \left(-1 \cdot \left(c\_p \cdot \log 2\right) + c\_p \cdot \log 2\right)}} \]

10. Simplified 99.5%

    \[\leadsto e^{\color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \left(c\_p + c\_n\right) \cdot \mathsf{fma}\left(t \cdot t, -0.005208333333333333, 0.125\right), 0.5 \cdot \left(c\_n - c\_p\right)\right), s \cdot \left(-0.5 \cdot \left(c\_n - c\_p\right)\right)\right)}} \]

11. Taylor expanded in t around 0

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

    1. *-commutative N/A

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

    2. associate-*r* N/A

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

    3. *-commutative N/A

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

    4. lower-*.f64 N/A

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

    5. lower-*.f64 N/A

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

    6. lower--.f64 97.3

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

13. Simplified 97.3%

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

14. Final simplification 97.3%

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

Alternative 7: 94.1% accurate, 896.0× 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 91.0%

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

3. Taylor expanded in c_n around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-neg.f64 93.4

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

5. Simplified 93.4%

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

6. Taylor expanded in c_p around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 94.2%

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

Developer Target 1: 96.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024219 
(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))))