Harley's example

Percentage Accurate: 91.4% → 97.3%

Time: 50.4s

Alternatives: 8

Speedup: 896.0×

• could not determine a ground truth

  1. c_p = 1.247345452407549e-201
  2. c_n = 1.39975461585509e-236
  3. t = -6.7035909706130115e-133
  4. s = 6.671297550419721e+107

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: 91.4% 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: 97.3% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(c\_p, t \cdot -0.5, 1\right) \cdot {\left(\frac{2}{\mathsf{fma}\left(s, \mathsf{fma}\left(0.5, s, -1\right), 2\right)}\right)}^{c\_p} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.6%

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

3. Taylor expanded in c_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. /-lowering-/.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. pow-lowering-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. /-lowering-/.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. +-lowering-+.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. exp-lowering-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. neg-lowering-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. pow-lowering-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. /-lowering-/.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. +-lowering-+.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. exp-lowering-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. neg-lowering-neg.f64 92.2

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

5. Simplified 92.2%

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

   1. div-inv N/A

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

   2. pow-flip N/A

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

   3. inv-pow N/A

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

   4. pow-unpow N/A

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

   5. neg-mul-1 N/A

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

   6. remove-double-neg N/A

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

   7. pow-prod-down N/A

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

   8. pow-lowering-pow.f64 N/A

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

7. Applied egg-rr 92.5%

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

8. Taylor expanded in t around 0

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. distribute-lft1-in N/A

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

   4. +-commutative N/A

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

   5. *-lowering-*.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. *-lowering-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. associate-*r/ N/A

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

   15. pow-lowering-pow.f64 N/A

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

10. Simplified 93.9%

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

11. Taylor expanded in s around 0

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

    1. +-commutative N/A

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

    2. accelerator-lowering-fma.f64 N/A

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

    3. sub-neg N/A

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

    4. metadata-eval N/A

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

    5. accelerator-lowering-fma.f64 97.3

       \[\leadsto \mathsf{fma}\left(c\_p, t \cdot -0.5, 1\right) \cdot {\left(\frac{2}{\mathsf{fma}\left(s, \color{blue}{\mathsf{fma}\left(0.5, s, -1\right)}, 2\right)}\right)}^{c\_p} \]

13. Simplified 97.3%

    \[\leadsto \mathsf{fma}\left(c\_p, t \cdot -0.5, 1\right) \cdot {\left(\frac{2}{\color{blue}{\mathsf{fma}\left(s, \mathsf{fma}\left(0.5, s, -1\right), 2\right)}}\right)}^{c\_p} \]
14. Add Preprocessing

Alternative 2: 95.4% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(c\_p, t \cdot -0.5, 1\right) \cdot {\left(\frac{2}{2 - s}\right)}^{c\_p} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.6%

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

3. Taylor expanded in c_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. /-lowering-/.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. pow-lowering-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. /-lowering-/.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. +-lowering-+.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. exp-lowering-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. neg-lowering-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. pow-lowering-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. /-lowering-/.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. +-lowering-+.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. exp-lowering-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. neg-lowering-neg.f64 92.2

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

5. Simplified 92.2%

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

   1. div-inv N/A

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

   2. pow-flip N/A

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

   3. inv-pow N/A

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

   4. pow-unpow N/A

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

   5. neg-mul-1 N/A

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

   6. remove-double-neg N/A

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

   7. pow-prod-down N/A

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

   8. pow-lowering-pow.f64 N/A

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

7. Applied egg-rr 92.5%

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

8. Taylor expanded in t around 0

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. distribute-lft1-in N/A

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

   4. +-commutative N/A

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

   5. *-lowering-*.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. *-lowering-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. associate-*r/ N/A

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

   15. pow-lowering-pow.f64 N/A

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

10. Simplified 93.9%

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

11. Taylor expanded in s around 0

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

    1. neg-mul-1 N/A

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

    2. unsub-neg N/A

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

    3. --lowering--.f64 94.5

       \[\leadsto \mathsf{fma}\left(c\_p, t \cdot -0.5, 1\right) \cdot {\left(\frac{2}{\color{blue}{2 - s}}\right)}^{c\_p} \]

13. Simplified 94.5%

    \[\leadsto \mathsf{fma}\left(c\_p, t \cdot -0.5, 1\right) \cdot {\left(\frac{2}{\color{blue}{2 - s}}\right)}^{c\_p} \]
14. Add Preprocessing

Alternative 3: 94.5% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(c\_p, t \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(s, \mathsf{fma}\left(s, \mathsf{fma}\left(s, \left(c\_p \cdot c\_p\right) \cdot \mathsf{fma}\left(c\_p, 0.020833333333333332, -0.0625\right), c\_p \cdot \mathsf{fma}\left(0.125, c\_p, -0.125\right)\right), c\_p \cdot 0.5\right), 1\right) \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (*
  (fma c_p (* t -0.5) 1.0)
  (fma
   s
   (fma
    s
    (fma
     s
     (* (* c_p c_p) (fma c_p 0.020833333333333332 -0.0625))
     (* c_p (fma 0.125 c_p -0.125)))
    (* c_p 0.5))
   1.0)))

C

double code(double c_p, double c_n, double t, double s) {
	return fma(c_p, (t * -0.5), 1.0) * fma(s, fma(s, fma(s, ((c_p * c_p) * fma(c_p, 0.020833333333333332, -0.0625)), (c_p * fma(0.125, c_p, -0.125))), (c_p * 0.5)), 1.0);
}

Julia

function code(c_p, c_n, t, s)
	return Float64(fma(c_p, Float64(t * -0.5), 1.0) * fma(s, fma(s, fma(s, Float64(Float64(c_p * c_p) * fma(c_p, 0.020833333333333332, -0.0625)), Float64(c_p * fma(0.125, c_p, -0.125))), Float64(c_p * 0.5)), 1.0))
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[(N[(c$95$p * N[(t * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(s * N[(s * N[(s * N[(N[(c$95$p * c$95$p), $MachinePrecision] * N[(c$95$p * 0.020833333333333332 + -0.0625), $MachinePrecision]), $MachinePrecision] + N[(c$95$p * N[(0.125 * c$95$p + -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c$95$p * 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.6%

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

3. Taylor expanded in c_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. /-lowering-/.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. pow-lowering-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. /-lowering-/.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. +-lowering-+.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. exp-lowering-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. neg-lowering-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. pow-lowering-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. /-lowering-/.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. +-lowering-+.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. exp-lowering-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. neg-lowering-neg.f64 92.2

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

5. Simplified 92.2%

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

   1. div-inv N/A

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

   2. pow-flip N/A

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

   3. inv-pow N/A

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

   4. pow-unpow N/A

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

   5. neg-mul-1 N/A

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

   6. remove-double-neg N/A

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

   7. pow-prod-down N/A

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

   8. pow-lowering-pow.f64 N/A

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

7. Applied egg-rr 92.5%

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

8. Taylor expanded in t around 0

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. distribute-lft1-in N/A

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

   4. +-commutative N/A

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

   5. *-lowering-*.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. *-lowering-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. associate-*r/ N/A

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

   15. pow-lowering-pow.f64 N/A

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

10. Simplified 93.9%

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

11. Taylor expanded in s around 0

    \[\leadsto \mathsf{fma}\left(c\_p, t \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\left(1 + s \cdot \left(\frac{1}{2} \cdot c\_p + s \cdot \left(\frac{-1}{8} \cdot c\_p + \left(\frac{1}{8} \cdot {c\_p}^{2} + s \cdot \left(\frac{-1}{16} \cdot {c\_p}^{2} + \frac{1}{48} \cdot {c\_p}^{3}\right)\right)\right)\right)\right)} \]
12. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(c\_p, t \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\left(s \cdot \left(\frac{1}{2} \cdot c\_p + s \cdot \left(\frac{-1}{8} \cdot c\_p + \left(\frac{1}{8} \cdot {c\_p}^{2} + s \cdot \left(\frac{-1}{16} \cdot {c\_p}^{2} + \frac{1}{48} \cdot {c\_p}^{3}\right)\right)\right)\right) + 1\right)} \]

    2. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(c\_p, t \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\mathsf{fma}\left(s, \frac{1}{2} \cdot c\_p + s \cdot \left(\frac{-1}{8} \cdot c\_p + \left(\frac{1}{8} \cdot {c\_p}^{2} + s \cdot \left(\frac{-1}{16} \cdot {c\_p}^{2} + \frac{1}{48} \cdot {c\_p}^{3}\right)\right)\right), 1\right)} \]

13. Simplified 93.9%

    \[\leadsto \mathsf{fma}\left(c\_p, t \cdot -0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(s, \mathsf{fma}\left(s, \mathsf{fma}\left(s, \left(c\_p \cdot c\_p\right) \cdot \mathsf{fma}\left(c\_p, 0.020833333333333332, -0.0625\right), c\_p \cdot \mathsf{fma}\left(0.125, c\_p, -0.125\right)\right), 0.5 \cdot c\_p\right), 1\right)} \]

14. Final simplification 93.9%

    \[\leadsto \mathsf{fma}\left(c\_p, t \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(s, \mathsf{fma}\left(s, \mathsf{fma}\left(s, \left(c\_p \cdot c\_p\right) \cdot \mathsf{fma}\left(c\_p, 0.020833333333333332, -0.0625\right), c\_p \cdot \mathsf{fma}\left(0.125, c\_p, -0.125\right)\right), c\_p \cdot 0.5\right), 1\right) \]
15. Add Preprocessing

Alternative 4: 94.6% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(0.125, \mathsf{fma}\left(c\_p, c\_p, c\_p\right), t \cdot \left(\left(c\_p \cdot c\_p\right) \cdot \mathsf{fma}\left(c\_p, -0.020833333333333332, -0.0625\right)\right)\right), c\_p \cdot -0.5\right), 1\right) \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (fma
  t
  (fma
   t
   (fma
    0.125
    (fma c_p c_p c_p)
    (* t (* (* c_p c_p) (fma c_p -0.020833333333333332 -0.0625))))
   (* c_p -0.5))
  1.0))

C

double code(double c_p, double c_n, double t, double s) {
	return fma(t, fma(t, fma(0.125, fma(c_p, c_p, c_p), (t * ((c_p * c_p) * fma(c_p, -0.020833333333333332, -0.0625)))), (c_p * -0.5)), 1.0);
}

Julia

function code(c_p, c_n, t, s)
	return fma(t, fma(t, fma(0.125, fma(c_p, c_p, c_p), Float64(t * Float64(Float64(c_p * c_p) * fma(c_p, -0.020833333333333332, -0.0625)))), Float64(c_p * -0.5)), 1.0)
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[(t * N[(t * N[(0.125 * N[(c$95$p * c$95$p + c$95$p), $MachinePrecision] + N[(t * N[(N[(c$95$p * c$95$p), $MachinePrecision] * N[(c$95$p * -0.020833333333333332 + -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c$95$p * -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 92.6%

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

3. Taylor expanded in c_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. /-lowering-/.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. pow-lowering-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. /-lowering-/.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. +-lowering-+.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. exp-lowering-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. neg-lowering-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. pow-lowering-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. /-lowering-/.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. +-lowering-+.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. exp-lowering-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. neg-lowering-neg.f64 92.2

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

5. Simplified 92.2%

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

   1. div-inv N/A

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

   2. pow-flip N/A

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

   3. inv-pow N/A

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

   4. pow-unpow N/A

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

   5. neg-mul-1 N/A

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

   6. remove-double-neg N/A

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

   7. pow-prod-down N/A

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

   8. pow-lowering-pow.f64 N/A

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

7. Applied egg-rr 92.5%

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

8. Taylor expanded in s around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. metadata-eval N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. neg-mul-1 N/A

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

   6. exp-lowering-exp.f64 N/A

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

   7. neg-mul-1 N/A

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

   8. neg-lowering-neg.f64 92.7

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

10. Simplified 92.7%

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

11. Taylor expanded in t around 0

    \[\leadsto \color{blue}{1 + t \cdot \left(\frac{-1}{2} \cdot c\_p + t \cdot \left(\frac{1}{8} \cdot c\_p + \left(\frac{1}{8} \cdot {c\_p}^{2} + t \cdot \left(\frac{-1}{16} \cdot {c\_p}^{2} + \frac{-1}{48} \cdot {c\_p}^{3}\right)\right)\right)\right)} \]
12. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \color{blue}{t \cdot \left(\frac{-1}{2} \cdot c\_p + t \cdot \left(\frac{1}{8} \cdot c\_p + \left(\frac{1}{8} \cdot {c\_p}^{2} + t \cdot \left(\frac{-1}{16} \cdot {c\_p}^{2} + \frac{-1}{48} \cdot {c\_p}^{3}\right)\right)\right)\right) + 1} \]

    2. accelerator-lowering-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(t, \frac{-1}{2} \cdot c\_p + t \cdot \left(\frac{1}{8} \cdot c\_p + \left(\frac{1}{8} \cdot {c\_p}^{2} + t \cdot \left(\frac{-1}{16} \cdot {c\_p}^{2} + \frac{-1}{48} \cdot {c\_p}^{3}\right)\right)\right), 1\right)} \]

13. Simplified 93.8%

    \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(0.125, \mathsf{fma}\left(c\_p, c\_p, c\_p\right), t \cdot \left(\left(c\_p \cdot c\_p\right) \cdot \mathsf{fma}\left(c\_p, -0.020833333333333332, -0.0625\right)\right)\right), -0.5 \cdot c\_p\right), 1\right)} \]

14. Final simplification 93.8%

    \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(0.125, \mathsf{fma}\left(c\_p, c\_p, c\_p\right), t \cdot \left(\left(c\_p \cdot c\_p\right) \cdot \mathsf{fma}\left(c\_p, -0.020833333333333332, -0.0625\right)\right)\right), c\_p \cdot -0.5\right), 1\right) \]
15. Add Preprocessing

Alternative 5: 94.5% accurate, 19.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(c\_p, t \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(s, \mathsf{fma}\left(0.5, c\_p, s \cdot \left(c\_p \cdot \mathsf{fma}\left(0.125, c\_p, -0.125\right)\right)\right), 1\right) \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (*
  (fma c_p (* t -0.5) 1.0)
  (fma s (fma 0.5 c_p (* s (* c_p (fma 0.125 c_p -0.125)))) 1.0)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.6%

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

3. Taylor expanded in c_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. /-lowering-/.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. pow-lowering-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. /-lowering-/.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. +-lowering-+.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. exp-lowering-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. neg-lowering-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. pow-lowering-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. /-lowering-/.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. +-lowering-+.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. exp-lowering-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. neg-lowering-neg.f64 92.2

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

5. Simplified 92.2%

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

   1. div-inv N/A

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

   2. pow-flip N/A

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

   3. inv-pow N/A

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

   4. pow-unpow N/A

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

   5. neg-mul-1 N/A

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

   6. remove-double-neg N/A

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

   7. pow-prod-down N/A

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

   8. pow-lowering-pow.f64 N/A

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

7. Applied egg-rr 92.5%

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

8. Taylor expanded in t around 0

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. distribute-lft1-in N/A

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

   4. +-commutative N/A

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

   5. *-lowering-*.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. *-lowering-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. associate-*r/ N/A

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

   15. pow-lowering-pow.f64 N/A

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

10. Simplified 93.9%

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

11. Taylor expanded in s around 0

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

    1. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(c\_p, t \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\left(s \cdot \left(\frac{1}{2} \cdot c\_p + s \cdot \left(\frac{-1}{8} \cdot c\_p + \frac{1}{8} \cdot {c\_p}^{2}\right)\right) + 1\right)} \]

    2. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(c\_p, t \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\mathsf{fma}\left(s, \frac{1}{2} \cdot c\_p + s \cdot \left(\frac{-1}{8} \cdot c\_p + \frac{1}{8} \cdot {c\_p}^{2}\right), 1\right)} \]

    3. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(c\_p, t \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(s, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, c\_p, s \cdot \left(\frac{-1}{8} \cdot c\_p + \frac{1}{8} \cdot {c\_p}^{2}\right)\right)}, 1\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(c\_p, t \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(s, \mathsf{fma}\left(\frac{1}{2}, c\_p, \color{blue}{s \cdot \left(\frac{-1}{8} \cdot c\_p + \frac{1}{8} \cdot {c\_p}^{2}\right)}\right), 1\right) \]

    5. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(c\_p, t \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(s, \mathsf{fma}\left(\frac{1}{2}, c\_p, s \cdot \color{blue}{\left(\frac{1}{8} \cdot {c\_p}^{2} + \frac{-1}{8} \cdot c\_p\right)}\right), 1\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(c\_p, t \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(s, \mathsf{fma}\left(\frac{1}{2}, c\_p, s \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(c\_p \cdot c\_p\right)} + \frac{-1}{8} \cdot c\_p\right)\right), 1\right) \]

    7. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(c\_p, t \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(s, \mathsf{fma}\left(\frac{1}{2}, c\_p, s \cdot \left(\color{blue}{\left(\frac{1}{8} \cdot c\_p\right) \cdot c\_p} + \frac{-1}{8} \cdot c\_p\right)\right), 1\right) \]

    8. distribute-rgt-out N/A

       \[\leadsto \mathsf{fma}\left(c\_p, t \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(s, \mathsf{fma}\left(\frac{1}{2}, c\_p, s \cdot \color{blue}{\left(c\_p \cdot \left(\frac{1}{8} \cdot c\_p + \frac{-1}{8}\right)\right)}\right), 1\right) \]

    9. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(c\_p, t \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(s, \mathsf{fma}\left(\frac{1}{2}, c\_p, s \cdot \color{blue}{\left(c\_p \cdot \left(\frac{1}{8} \cdot c\_p + \frac{-1}{8}\right)\right)}\right), 1\right) \]

    10. accelerator-lowering-fma.f64 93.8

        \[\leadsto \mathsf{fma}\left(c\_p, t \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(s, \mathsf{fma}\left(0.5, c\_p, s \cdot \left(c\_p \cdot \color{blue}{\mathsf{fma}\left(0.125, c\_p, -0.125\right)}\right)\right), 1\right) \]

13. Simplified 93.8%

    \[\leadsto \mathsf{fma}\left(c\_p, t \cdot -0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(s, \mathsf{fma}\left(0.5, c\_p, s \cdot \left(c\_p \cdot \mathsf{fma}\left(0.125, c\_p, -0.125\right)\right)\right), 1\right)} \]
14. Add Preprocessing

Alternative 6: 94.6% accurate, 37.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(t, c\_p \cdot \mathsf{fma}\left(0.125, \mathsf{fma}\left(c\_p, t, t\right), -0.5\right), 1\right) \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (fma t (* c_p (fma 0.125 (fma c_p t t) -0.5)) 1.0))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.6%

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

3. Taylor expanded in c_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. /-lowering-/.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. pow-lowering-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. /-lowering-/.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. +-lowering-+.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. exp-lowering-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. neg-lowering-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. pow-lowering-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. /-lowering-/.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. +-lowering-+.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. exp-lowering-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. neg-lowering-neg.f64 92.2

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

5. Simplified 92.2%

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

   1. div-inv N/A

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

   2. pow-flip N/A

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

   3. inv-pow N/A

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

   4. pow-unpow N/A

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

   5. neg-mul-1 N/A

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

   6. remove-double-neg N/A

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

   7. pow-prod-down N/A

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

   8. pow-lowering-pow.f64 N/A

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

7. Applied egg-rr 92.5%

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

8. Taylor expanded in s around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. metadata-eval N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. neg-mul-1 N/A

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

   6. exp-lowering-exp.f64 N/A

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

   7. neg-mul-1 N/A

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

   8. neg-lowering-neg.f64 92.7

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

10. Simplified 92.7%

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

11. Taylor expanded in t around 0

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

    1. distribute-lft-in N/A

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

    2. associate-+r+ N/A

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

    3. distribute-lft-in N/A

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

    4. associate-*r* N/A

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

    5. *-commutative N/A

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

    6. associate-*r* N/A

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

    7. associate-+r+ N/A

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

    8. distribute-lft-in N/A

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

    9. +-commutative N/A

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

    10. accelerator-lowering-fma.f64 N/A

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

13. Simplified 93.8%

    \[\leadsto \color{blue}{\mathsf{fma}\left(t, c\_p \cdot \mathsf{fma}\left(0.125, \mathsf{fma}\left(c\_p, t, t\right), -0.5\right), 1\right)} \]
14. Add Preprocessing

Alternative 7: 94.5% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.6%

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

3. Taylor expanded in c_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. /-lowering-/.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. pow-lowering-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. /-lowering-/.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. +-lowering-+.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. exp-lowering-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. neg-lowering-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. pow-lowering-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. /-lowering-/.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. +-lowering-+.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. exp-lowering-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. neg-lowering-neg.f64 92.2

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

5. Simplified 92.2%

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

   1. div-inv N/A

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

   2. pow-flip N/A

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

   3. inv-pow N/A

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

   4. pow-unpow N/A

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

   5. neg-mul-1 N/A

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

   6. remove-double-neg N/A

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

   7. pow-prod-down N/A

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

   8. pow-lowering-pow.f64 N/A

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

7. Applied egg-rr 92.5%

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

8. Taylor expanded in s around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. metadata-eval N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. neg-mul-1 N/A

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

   6. exp-lowering-exp.f64 N/A

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

   7. neg-mul-1 N/A

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

   8. neg-lowering-neg.f64 92.7

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

10. Simplified 92.7%

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

11. Taylor expanded in t around 0

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

    1. +-commutative N/A

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

    2. accelerator-lowering-fma.f64 N/A

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

    3. *-lowering-*.f64 93.7

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

13. Simplified 93.7%

    \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, c\_p \cdot t, 1\right)} \]
14. Add Preprocessing

Alternative 8: 94.5% 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 92.6%

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

3. Taylor expanded in c_p around 0

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

   1. /-lowering-/.f64 N/A

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

   2. pow-lowering-pow.f64 N/A

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

   3. sub-neg N/A

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

   4. +-lowering-+.f64 N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. /-lowering-/.f64 N/A

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

   8. +-lowering-+.f64 N/A

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

   9. exp-lowering-exp.f64 N/A

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

   10. neg-lowering-neg.f64 N/A

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

   11. pow-lowering-pow.f64 N/A

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

5. Simplified 95.9%

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

6. Taylor expanded in c_n around 0

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

8. Simplified 93.5%

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

Developer Target 1: 96.8% 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 2024198 
(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))))