Harley's example

Percentage Accurate: 90.9% → 95.9%

Time: 1.4min

Alternatives: 8

Speedup: 896.0×

• could not determine a ground truth

  1. c_p = 1.460040832169826e+212
  2. c_n = 7.792922809436725e-35
  3. t = 2.389570559595535e+73
  4. s = -8.395999787141248e-199

Specification

\[0 < c\_p \land 0 < c\_n\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{1 + e^{-t}}\\ t_2 := \frac{1}{1 + e^{-s}}\\ \frac{{t\_2}^{c\_p} \cdot {\left(1 - t\_2\right)}^{c\_n}}{{t\_1}^{c\_p} \cdot {\left(1 - t\_1\right)}^{c\_n}} \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (/ 1.0 (+ 1.0 (exp (- t))))) (t_2 (/ 1.0 (+ 1.0 (exp (- s))))))
   (/
    (* (pow t_2 c_p) (pow (- 1.0 t_2) c_n))
    (* (pow t_1 c_p) (pow (- 1.0 t_1) c_n)))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = 1.0 / (1.0 + exp(-t));
	double t_2 = 1.0 / (1.0 + exp(-s));
	return (pow(t_2, c_p) * pow((1.0 - t_2), c_n)) / (pow(t_1, c_p) * pow((1.0 - t_1), c_n));
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    real(8) :: t_1
    real(8) :: t_2
    t_1 = 1.0d0 / (1.0d0 + exp(-t))
    t_2 = 1.0d0 / (1.0d0 + exp(-s))
    code = ((t_2 ** c_p) * ((1.0d0 - t_2) ** c_n)) / ((t_1 ** c_p) * ((1.0d0 - t_1) ** c_n))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double t_1 = 1.0 / (1.0 + Math.exp(-t));
	double t_2 = 1.0 / (1.0 + Math.exp(-s));
	return (Math.pow(t_2, c_p) * Math.pow((1.0 - t_2), c_n)) / (Math.pow(t_1, c_p) * Math.pow((1.0 - t_1), c_n));
}

Python

def code(c_p, c_n, t, s):
	t_1 = 1.0 / (1.0 + math.exp(-t))
	t_2 = 1.0 / (1.0 + math.exp(-s))
	return (math.pow(t_2, c_p) * math.pow((1.0 - t_2), c_n)) / (math.pow(t_1, c_p) * math.pow((1.0 - t_1), c_n))

Julia

function code(c_p, c_n, t, s)
	t_1 = Float64(1.0 / Float64(1.0 + exp(Float64(-t))))
	t_2 = Float64(1.0 / Float64(1.0 + exp(Float64(-s))))
	return Float64(Float64((t_2 ^ c_p) * (Float64(1.0 - t_2) ^ c_n)) / Float64((t_1 ^ c_p) * (Float64(1.0 - t_1) ^ c_n)))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	t_1 = 1.0 / (1.0 + exp(-t));
	t_2 = 1.0 / (1.0 + exp(-s));
	tmp = ((t_2 ^ c_p) * ((1.0 - t_2) ^ c_n)) / ((t_1 ^ c_p) * ((1.0 - t_1) ^ c_n));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[(1.0 / N[(1.0 + N[Exp[(-t)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(1.0 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[t$95$2, c$95$p], $MachinePrecision] * N[Power[N[(1.0 - t$95$2), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$1, c$95$p], $MachinePrecision] * N[Power[N[(1.0 - t$95$1), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Initial Program: 90.9% 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: 95.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ {\left(\left(1 + \frac{-1}{1 + e^{0 - s}}\right) \cdot 2\right)}^{c\_n} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (pow (* (+ 1.0 (/ -1.0 (+ 1.0 (exp (- 0.0 s))))) 2.0) c_n))

C

double code(double c_p, double c_n, double t, double s) {
	return pow(((1.0 + (-1.0 / (1.0 + exp((0.0 - s))))) * 2.0), 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 + ((-1.0d0) / (1.0d0 + exp((0.0d0 - s))))) * 2.0d0) ** c_n
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return Math.pow(((1.0 + (-1.0 / (1.0 + Math.exp((0.0 - s))))) * 2.0), c_n);
}

Python

def code(c_p, c_n, t, s):
	return math.pow(((1.0 + (-1.0 / (1.0 + math.exp((0.0 - s))))) * 2.0), c_n)

Julia

function code(c_p, c_n, t, s)
	return Float64(Float64(1.0 + Float64(-1.0 / Float64(1.0 + exp(Float64(0.0 - s))))) * 2.0) ^ c_n
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = ((1.0 + (-1.0 / (1.0 + exp((0.0 - s))))) * 2.0) ^ c_n;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[Power[N[(N[(1.0 + N[(-1.0 / N[(1.0 + N[Exp[N[(0.0 - s), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], c$95$n], $MachinePrecision]

Derivation

1. Initial program 90.7%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
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-sub0 N/A

       \[\leadsto \frac{{\left(1 + \frac{-1}{1 + e^{\color{blue}{0 - s}}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

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

       \[\leadsto \frac{{\left(1 + \frac{-1}{1 + e^{\color{blue}{0 - s}}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

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

       \[\leadsto \frac{{\left(1 + \frac{-1}{1 + e^{0 - s}}\right)}^{c\_n}}{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

5. Simplified 93.8%

   \[\leadsto \color{blue}{\frac{{\left(1 + \frac{-1}{1 + e^{0 - s}}\right)}^{c\_n}}{{\left(1 + \frac{-1}{1 + e^{0 - t}}\right)}^{c\_n}}} \]

6. Taylor expanded in t around 0

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

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

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

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

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

   3. sub-neg N/A

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

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

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

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

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

   8. neg-mul-1 N/A

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

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

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

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

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

   11. neg-mul-1 N/A

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

   12. neg-sub0 N/A

       \[\leadsto \frac{{\left(1 + \frac{-1}{1 + e^{\color{blue}{0 - s}}}\right)}^{c\_n}}{{\frac{1}{2}}^{c\_n}} \]

   13. --lowering--.f64 N/A

       \[\leadsto \frac{{\left(1 + \frac{-1}{1 + e^{\color{blue}{0 - s}}}\right)}^{c\_n}}{{\frac{1}{2}}^{c\_n}} \]

   14. pow-lowering-pow.f64 93.8

       \[\leadsto \frac{{\left(1 + \frac{-1}{1 + e^{0 - s}}\right)}^{c\_n}}{\color{blue}{{0.5}^{c\_n}}} \]

8. Simplified 93.8%

   \[\leadsto \color{blue}{\frac{{\left(1 + \frac{-1}{1 + e^{0 - s}}\right)}^{c\_n}}{{0.5}^{c\_n}}} \]
9. Step-by-step derivation

   1. div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. pow-flip N/A

      \[\leadsto {\left(1 + \frac{-1}{1 + e^{0 - s}}\right)}^{c\_n} \cdot \color{blue}{{\frac{1}{2}}^{\left(\mathsf{neg}\left(c\_n\right)\right)}} \]

   10. neg-mul-1 N/A

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

   11. pow-unpow N/A

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

   12. metadata-eval N/A

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

   13. pow-lowering-pow.f64 93.7

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

10. Applied egg-rr 93.7%

    \[\leadsto \color{blue}{{\left(1 + \frac{-1}{1 + e^{0 - s}}\right)}^{c\_n} \cdot {2}^{c\_n}} \]
11. Step-by-step derivation

    1. pow-prod-down N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

    8. --lowering--.f64 97.7

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

12. Applied egg-rr 97.7%

    \[\leadsto \color{blue}{{\left(\left(1 + \frac{-1}{1 + e^{0 - s}}\right) \cdot 2\right)}^{c\_n}} \]
13. Add Preprocessing

Alternative 2: 95.5% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(0 - t, 0.5 \cdot c\_p, 1\right) + t \cdot \left(c\_n \cdot -0.5\right)\\ \mathbf{if}\;0 - s \leq -5 \cdot 10^{-12}:\\ \;\;\;\;{0.5}^{c\_n}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0 - t, \mathsf{fma}\left(c\_p, 0.5, c\_n \cdot -0.5\right), 1\right) \cdot t\_1\right) \cdot \frac{1}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (+ (fma (- 0.0 t) (* 0.5 c_p) 1.0) (* t (* c_n -0.5)))))
   (if (<= (- 0.0 s) -5e-12)
     (pow 0.5 c_n)
     (* (* (fma (- 0.0 t) (fma c_p 0.5 (* c_n -0.5)) 1.0) t_1) (/ 1.0 t_1)))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = fma((0.0 - t), (0.5 * c_p), 1.0) + (t * (c_n * -0.5));
	double tmp;
	if ((0.0 - s) <= -5e-12) {
		tmp = pow(0.5, c_n);
	} else {
		tmp = (fma((0.0 - t), fma(c_p, 0.5, (c_n * -0.5)), 1.0) * t_1) * (1.0 / t_1);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	t_1 = Float64(fma(Float64(0.0 - t), Float64(0.5 * c_p), 1.0) + Float64(t * Float64(c_n * -0.5)))
	tmp = 0.0
	if (Float64(0.0 - s) <= -5e-12)
		tmp = 0.5 ^ c_n;
	else
		tmp = Float64(Float64(fma(Float64(0.0 - t), fma(c_p, 0.5, Float64(c_n * -0.5)), 1.0) * t_1) * Float64(1.0 / t_1));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[(N[(N[(0.0 - t), $MachinePrecision] * N[(0.5 * c$95$p), $MachinePrecision] + 1.0), $MachinePrecision] + N[(t * N[(c$95$n * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(0.0 - s), $MachinePrecision], -5e-12], N[Power[0.5, c$95$n], $MachinePrecision], N[(N[(N[(N[(0.0 - t), $MachinePrecision] * N[(c$95$p * 0.5 + N[(c$95$n * -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 s) < -4.9999999999999997e-12

   1. Initial program 47.2%

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

   3. Taylor expanded in c_n around 0

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

   5. Simplified 82.5%

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

   6. Taylor expanded in c_p around 0

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

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

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

      2. sub-neg N/A

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

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

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

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

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

      7. neg-mul-1 N/A

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

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

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

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

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

      10. neg-mul-1 N/A

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

      11. neg-sub0 N/A

          \[\leadsto {\left(1 + \frac{-1}{1 + e^{\color{blue}{0 - s}}}\right)}^{c\_n} \]

      12. --lowering--.f64 94.2

          \[\leadsto {\left(1 + \frac{-1}{1 + e^{\color{blue}{0 - s}}}\right)}^{c\_n} \]

   8. Simplified 94.2%

      \[\leadsto \color{blue}{{\left(1 + \frac{-1}{1 + e^{0 - s}}\right)}^{c\_n}} \]

   9. Taylor expanded in s around 0

      \[\leadsto \color{blue}{{\frac{1}{2}}^{c\_n}} \]
   10. Step-by-step derivation

       1. pow-lowering-pow.f64 94.3

          \[\leadsto \color{blue}{{0.5}^{c\_n}} \]

   11. Simplified 94.3%

       \[\leadsto \color{blue}{{0.5}^{c\_n}} \]

   if -4.9999999999999997e-12 < (neg.f64 s)

   1. Initial program 93.8%

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

   3. Taylor expanded in s around 0

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

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

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

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

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

      3. pow-lowering-pow.f64 94.7

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

   5. Simplified 94.7%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. neg-mul-1 N/A

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

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

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

      5. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - t}, \frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - t}, \frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(0 - t, \color{blue}{\frac{1}{2} \cdot c\_p + \frac{-1}{2} \cdot c\_n}, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(0 - t, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, c\_p, \frac{-1}{2} \cdot c\_n\right)}, 1\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(0 - t, \mathsf{fma}\left(\frac{1}{2}, c\_p, \color{blue}{c\_n \cdot \frac{-1}{2}}\right), 1\right) \]

      10. *-lowering-*.f64 97.5

          \[\leadsto \mathsf{fma}\left(0 - t, \mathsf{fma}\left(0.5, c\_p, \color{blue}{c\_n \cdot -0.5}\right), 1\right) \]

   8. Simplified 97.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0 - t, \mathsf{fma}\left(0.5, c\_p, c\_n \cdot -0.5\right), 1\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+r+ N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

      13. sub0-neg N/A

          \[\leadsto \left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) + \color{blue}{\left(\mathsf{neg}\left(t\right)\right)} \cdot \left(c\_n \cdot \frac{-1}{2}\right) \]

      14. distribute-lft-neg-out N/A

          \[\leadsto \left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) + \color{blue}{\left(\mathsf{neg}\left(t \cdot \left(c\_n \cdot \frac{-1}{2}\right)\right)\right)} \]

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

          \[\leadsto \left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) + \color{blue}{\left(\mathsf{neg}\left(t \cdot \left(c\_n \cdot \frac{-1}{2}\right)\right)\right)} \]

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

          \[\leadsto \left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) + \left(\mathsf{neg}\left(\color{blue}{t \cdot \left(c\_n \cdot \frac{-1}{2}\right)}\right)\right) \]

      17. *-lowering-*.f64 97.5

          \[\leadsto \left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot 0.5\right) + \left(-t \cdot \color{blue}{\left(c\_n \cdot -0.5\right)}\right) \]

   10. Applied egg-rr 97.5%

       \[\leadsto \color{blue}{\left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot 0.5\right) + \left(-t \cdot \left(c\_n \cdot -0.5\right)\right)} \]
   11. Step-by-step derivation

       1. flip-+ N/A

          \[\leadsto \color{blue}{\frac{\left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) \cdot \left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) - \left(\mathsf{neg}\left(t \cdot \left(c\_n \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(\mathsf{neg}\left(t \cdot \left(c\_n \cdot \frac{-1}{2}\right)\right)\right)}{\left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) - \left(\mathsf{neg}\left(t \cdot \left(c\_n \cdot \frac{-1}{2}\right)\right)\right)}} \]

       2. div-inv N/A

          \[\leadsto \color{blue}{\left(\left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) \cdot \left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) - \left(\mathsf{neg}\left(t \cdot \left(c\_n \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(\mathsf{neg}\left(t \cdot \left(c\_n \cdot \frac{-1}{2}\right)\right)\right)\right) \cdot \frac{1}{\left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) - \left(\mathsf{neg}\left(t \cdot \left(c\_n \cdot \frac{-1}{2}\right)\right)\right)}} \]

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

          \[\leadsto \color{blue}{\left(\left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) \cdot \left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) - \left(\mathsf{neg}\left(t \cdot \left(c\_n \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(\mathsf{neg}\left(t \cdot \left(c\_n \cdot \frac{-1}{2}\right)\right)\right)\right) \cdot \frac{1}{\left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) - \left(\mathsf{neg}\left(t \cdot \left(c\_n \cdot \frac{-1}{2}\right)\right)\right)}} \]

   12. Applied egg-rr 97.5%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0 - t, \mathsf{fma}\left(c\_p, 0.5, c\_n \cdot -0.5\right), 1\right) \cdot \left(\mathsf{fma}\left(0 - t, c\_p \cdot 0.5, 1\right) + t \cdot \left(c\_n \cdot -0.5\right)\right)\right) \cdot \frac{1}{\mathsf{fma}\left(0 - t, c\_p \cdot 0.5, 1\right) + t \cdot \left(c\_n \cdot -0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0 - s \leq -5 \cdot 10^{-12}:\\ \;\;\;\;{0.5}^{c\_n}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0 - t, \mathsf{fma}\left(c\_p, 0.5, c\_n \cdot -0.5\right), 1\right) \cdot \left(\mathsf{fma}\left(0 - t, 0.5 \cdot c\_p, 1\right) + t \cdot \left(c\_n \cdot -0.5\right)\right)\right) \cdot \frac{1}{\mathsf{fma}\left(0 - t, 0.5 \cdot c\_p, 1\right) + t \cdot \left(c\_n \cdot -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.8% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(0 - t, 0.5 \cdot c\_p, 1\right) + t \cdot \left(c\_n \cdot -0.5\right)\\ \mathbf{if}\;0 - s \leq -1 \cdot 10^{+38}:\\ \;\;\;\;t \cdot \left(c\_p \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0 - t, \mathsf{fma}\left(c\_p, 0.5, c\_n \cdot -0.5\right), 1\right) \cdot t\_1\right) \cdot \frac{1}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (+ (fma (- 0.0 t) (* 0.5 c_p) 1.0) (* t (* c_n -0.5)))))
   (if (<= (- 0.0 s) -1e+38)
     (* t (* c_p -0.5))
     (* (* (fma (- 0.0 t) (fma c_p 0.5 (* c_n -0.5)) 1.0) t_1) (/ 1.0 t_1)))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = fma((0.0 - t), (0.5 * c_p), 1.0) + (t * (c_n * -0.5));
	double tmp;
	if ((0.0 - s) <= -1e+38) {
		tmp = t * (c_p * -0.5);
	} else {
		tmp = (fma((0.0 - t), fma(c_p, 0.5, (c_n * -0.5)), 1.0) * t_1) * (1.0 / t_1);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	t_1 = Float64(fma(Float64(0.0 - t), Float64(0.5 * c_p), 1.0) + Float64(t * Float64(c_n * -0.5)))
	tmp = 0.0
	if (Float64(0.0 - s) <= -1e+38)
		tmp = Float64(t * Float64(c_p * -0.5));
	else
		tmp = Float64(Float64(fma(Float64(0.0 - t), fma(c_p, 0.5, Float64(c_n * -0.5)), 1.0) * t_1) * Float64(1.0 / t_1));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[(N[(N[(0.0 - t), $MachinePrecision] * N[(0.5 * c$95$p), $MachinePrecision] + 1.0), $MachinePrecision] + N[(t * N[(c$95$n * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(0.0 - s), $MachinePrecision], -1e+38], N[(t * N[(c$95$p * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.0 - t), $MachinePrecision] * N[(c$95$p * 0.5 + N[(c$95$n * -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 s) < -9.99999999999999977e37

   1. Initial program 0.0%

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

   3. Taylor expanded in s around 0

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

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

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

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

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

      3. pow-lowering-pow.f64 0.0

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

   5. Simplified 0.0%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. neg-mul-1 N/A

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

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

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

      5. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - t}, \frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - t}, \frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(0 - t, \color{blue}{\frac{1}{2} \cdot c\_p + \frac{-1}{2} \cdot c\_n}, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(0 - t, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, c\_p, \frac{-1}{2} \cdot c\_n\right)}, 1\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(0 - t, \mathsf{fma}\left(\frac{1}{2}, c\_p, \color{blue}{c\_n \cdot \frac{-1}{2}}\right), 1\right) \]

      10. *-lowering-*.f64 3.1

          \[\leadsto \mathsf{fma}\left(0 - t, \mathsf{fma}\left(0.5, c\_p, \color{blue}{c\_n \cdot -0.5}\right), 1\right) \]

   8. Simplified 3.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0 - t, \mathsf{fma}\left(0.5, c\_p, c\_n \cdot -0.5\right), 1\right)} \]

   9. Taylor expanded in c_p around inf

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

       1. associate-*r* N/A

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

       2. metadata-eval N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. associate-*r* N/A

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

       7. metadata-eval N/A

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

       8. *-lowering-*.f64 73.1

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

   11. Simplified 73.1%

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

   if -9.99999999999999977e37 < (neg.f64 s)

   1. Initial program 93.3%

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

   3. Taylor expanded in s around 0

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

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

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

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

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

      3. pow-lowering-pow.f64 94.1

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

   5. Simplified 94.1%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. neg-mul-1 N/A

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

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

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

      5. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - t}, \frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - t}, \frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(0 - t, \color{blue}{\frac{1}{2} \cdot c\_p + \frac{-1}{2} \cdot c\_n}, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(0 - t, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, c\_p, \frac{-1}{2} \cdot c\_n\right)}, 1\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(0 - t, \mathsf{fma}\left(\frac{1}{2}, c\_p, \color{blue}{c\_n \cdot \frac{-1}{2}}\right), 1\right) \]

      10. *-lowering-*.f64 97.2

          \[\leadsto \mathsf{fma}\left(0 - t, \mathsf{fma}\left(0.5, c\_p, \color{blue}{c\_n \cdot -0.5}\right), 1\right) \]

   8. Simplified 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0 - t, \mathsf{fma}\left(0.5, c\_p, c\_n \cdot -0.5\right), 1\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+r+ N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

      13. sub0-neg N/A

          \[\leadsto \left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) + \color{blue}{\left(\mathsf{neg}\left(t\right)\right)} \cdot \left(c\_n \cdot \frac{-1}{2}\right) \]

      14. distribute-lft-neg-out N/A

          \[\leadsto \left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) + \color{blue}{\left(\mathsf{neg}\left(t \cdot \left(c\_n \cdot \frac{-1}{2}\right)\right)\right)} \]

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

          \[\leadsto \left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) + \color{blue}{\left(\mathsf{neg}\left(t \cdot \left(c\_n \cdot \frac{-1}{2}\right)\right)\right)} \]

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

          \[\leadsto \left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) + \left(\mathsf{neg}\left(\color{blue}{t \cdot \left(c\_n \cdot \frac{-1}{2}\right)}\right)\right) \]

      17. *-lowering-*.f64 97.2

          \[\leadsto \left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot 0.5\right) + \left(-t \cdot \color{blue}{\left(c\_n \cdot -0.5\right)}\right) \]

   10. Applied egg-rr 97.2%

       \[\leadsto \color{blue}{\left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot 0.5\right) + \left(-t \cdot \left(c\_n \cdot -0.5\right)\right)} \]
   11. Step-by-step derivation

       1. flip-+ N/A

          \[\leadsto \color{blue}{\frac{\left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) \cdot \left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) - \left(\mathsf{neg}\left(t \cdot \left(c\_n \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(\mathsf{neg}\left(t \cdot \left(c\_n \cdot \frac{-1}{2}\right)\right)\right)}{\left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) - \left(\mathsf{neg}\left(t \cdot \left(c\_n \cdot \frac{-1}{2}\right)\right)\right)}} \]

       2. div-inv N/A

          \[\leadsto \color{blue}{\left(\left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) \cdot \left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) - \left(\mathsf{neg}\left(t \cdot \left(c\_n \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(\mathsf{neg}\left(t \cdot \left(c\_n \cdot \frac{-1}{2}\right)\right)\right)\right) \cdot \frac{1}{\left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) - \left(\mathsf{neg}\left(t \cdot \left(c\_n \cdot \frac{-1}{2}\right)\right)\right)}} \]

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

          \[\leadsto \color{blue}{\left(\left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) \cdot \left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) - \left(\mathsf{neg}\left(t \cdot \left(c\_n \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(\mathsf{neg}\left(t \cdot \left(c\_n \cdot \frac{-1}{2}\right)\right)\right)\right) \cdot \frac{1}{\left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) - \left(\mathsf{neg}\left(t \cdot \left(c\_n \cdot \frac{-1}{2}\right)\right)\right)}} \]

   12. Applied egg-rr 97.2%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0 - t, \mathsf{fma}\left(c\_p, 0.5, c\_n \cdot -0.5\right), 1\right) \cdot \left(\mathsf{fma}\left(0 - t, c\_p \cdot 0.5, 1\right) + t \cdot \left(c\_n \cdot -0.5\right)\right)\right) \cdot \frac{1}{\mathsf{fma}\left(0 - t, c\_p \cdot 0.5, 1\right) + t \cdot \left(c\_n \cdot -0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0 - s \leq -1 \cdot 10^{+38}:\\ \;\;\;\;t \cdot \left(c\_p \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0 - t, \mathsf{fma}\left(c\_p, 0.5, c\_n \cdot -0.5\right), 1\right) \cdot \left(\mathsf{fma}\left(0 - t, 0.5 \cdot c\_p, 1\right) + t \cdot \left(c\_n \cdot -0.5\right)\right)\right) \cdot \frac{1}{\mathsf{fma}\left(0 - t, 0.5 \cdot c\_p, 1\right) + t \cdot \left(c\_n \cdot -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.8% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 5600:\\ \;\;\;\;\mathsf{fma}\left(c\_n \cdot t, 0.5, \mathsf{fma}\left(0 - t, 0.5 \cdot c\_p, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(c\_p \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= s 5600.0)
   (fma (* c_n t) 0.5 (fma (- 0.0 t) (* 0.5 c_p) 1.0))
   (* t (* c_p -0.5))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (s <= 5600.0) {
		tmp = fma((c_n * t), 0.5, fma((0.0 - t), (0.5 * c_p), 1.0));
	} else {
		tmp = t * (c_p * -0.5);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (s <= 5600.0)
		tmp = fma(Float64(c_n * t), 0.5, fma(Float64(0.0 - t), Float64(0.5 * c_p), 1.0));
	else
		tmp = Float64(t * Float64(c_p * -0.5));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[s, 5600.0], N[(N[(c$95$n * t), $MachinePrecision] * 0.5 + N[(N[(0.0 - t), $MachinePrecision] * N[(0.5 * c$95$p), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t * N[(c$95$p * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < 5600

   1. Initial program 93.3%

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

   3. Taylor expanded in s around 0

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

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

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

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

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

      3. pow-lowering-pow.f64 94.1

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

   5. Simplified 94.1%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. neg-mul-1 N/A

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

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

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

      5. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - t}, \frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - t}, \frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(0 - t, \color{blue}{\frac{1}{2} \cdot c\_p + \frac{-1}{2} \cdot c\_n}, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(0 - t, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, c\_p, \frac{-1}{2} \cdot c\_n\right)}, 1\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(0 - t, \mathsf{fma}\left(\frac{1}{2}, c\_p, \color{blue}{c\_n \cdot \frac{-1}{2}}\right), 1\right) \]

      10. *-lowering-*.f64 97.2

          \[\leadsto \mathsf{fma}\left(0 - t, \mathsf{fma}\left(0.5, c\_p, \color{blue}{c\_n \cdot -0.5}\right), 1\right) \]

   8. Simplified 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0 - t, \mathsf{fma}\left(0.5, c\_p, c\_n \cdot -0.5\right), 1\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+r+ N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

      13. sub0-neg N/A

          \[\leadsto \left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) + \color{blue}{\left(\mathsf{neg}\left(t\right)\right)} \cdot \left(c\_n \cdot \frac{-1}{2}\right) \]

      14. distribute-lft-neg-out N/A

          \[\leadsto \left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) + \color{blue}{\left(\mathsf{neg}\left(t \cdot \left(c\_n \cdot \frac{-1}{2}\right)\right)\right)} \]

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

          \[\leadsto \left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) + \color{blue}{\left(\mathsf{neg}\left(t \cdot \left(c\_n \cdot \frac{-1}{2}\right)\right)\right)} \]

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

          \[\leadsto \left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) + \left(\mathsf{neg}\left(\color{blue}{t \cdot \left(c\_n \cdot \frac{-1}{2}\right)}\right)\right) \]

      17. *-lowering-*.f64 97.2

          \[\leadsto \left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot 0.5\right) + \left(-t \cdot \color{blue}{\left(c\_n \cdot -0.5\right)}\right) \]

   10. Applied egg-rr 97.2%

       \[\leadsto \color{blue}{\left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot 0.5\right) + \left(-t \cdot \left(c\_n \cdot -0.5\right)\right)} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t \cdot \left(c\_n \cdot \frac{-1}{2}\right)\right)\right) + \left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right)} \]

       2. associate-*r* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(t \cdot c\_n\right) \cdot \frac{-1}{2}}\right)\right) + \left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) \]

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

          \[\leadsto \color{blue}{\left(t \cdot c\_n\right) \cdot \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)} + \left(1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) \]

       4. metadata-eval N/A

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

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot c\_n, \frac{1}{2}, 1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right)} \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot c\_n}, \frac{1}{2}, 1 + \left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2}\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(t \cdot c\_n, \frac{1}{2}, \color{blue}{\left(\left(0 - t\right) \cdot c\_p\right) \cdot \frac{1}{2} + 1}\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(t \cdot c\_n, \frac{1}{2}, \color{blue}{\left(0 - t\right) \cdot \left(c\_p \cdot \frac{1}{2}\right)} + 1\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot c\_n, \frac{1}{2}, \left(0 - t\right) \cdot \left(c\_p \cdot \color{blue}{\frac{1}{2}}\right) + 1\right) \]

       10. div-inv N/A

           \[\leadsto \mathsf{fma}\left(t \cdot c\_n, \frac{1}{2}, \left(0 - t\right) \cdot \color{blue}{\frac{c\_p}{2}} + 1\right) \]

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

           \[\leadsto \mathsf{fma}\left(t \cdot c\_n, \frac{1}{2}, \color{blue}{\mathsf{fma}\left(0 - t, \frac{c\_p}{2}, 1\right)}\right) \]

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

           \[\leadsto \mathsf{fma}\left(t \cdot c\_n, \frac{1}{2}, \mathsf{fma}\left(\color{blue}{0 - t}, \frac{c\_p}{2}, 1\right)\right) \]

       13. div-inv N/A

           \[\leadsto \mathsf{fma}\left(t \cdot c\_n, \frac{1}{2}, \mathsf{fma}\left(0 - t, \color{blue}{c\_p \cdot \frac{1}{2}}, 1\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(t \cdot c\_n, \frac{1}{2}, \mathsf{fma}\left(0 - t, c\_p \cdot \color{blue}{\frac{1}{2}}, 1\right)\right) \]

       15. *-lowering-*.f64 97.2

           \[\leadsto \mathsf{fma}\left(t \cdot c\_n, 0.5, \mathsf{fma}\left(0 - t, \color{blue}{c\_p \cdot 0.5}, 1\right)\right) \]

   12. Applied egg-rr 97.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot c\_n, 0.5, \mathsf{fma}\left(0 - t, c\_p \cdot 0.5, 1\right)\right)} \]

   if 5600 < s

   1. Initial program 0.0%

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

   3. Taylor expanded in s around 0

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

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

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

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

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

      3. pow-lowering-pow.f64 0.0

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

   5. Simplified 0.0%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. neg-mul-1 N/A

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

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

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

      5. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - t}, \frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - t}, \frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(0 - t, \color{blue}{\frac{1}{2} \cdot c\_p + \frac{-1}{2} \cdot c\_n}, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(0 - t, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, c\_p, \frac{-1}{2} \cdot c\_n\right)}, 1\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(0 - t, \mathsf{fma}\left(\frac{1}{2}, c\_p, \color{blue}{c\_n \cdot \frac{-1}{2}}\right), 1\right) \]

      10. *-lowering-*.f64 3.1

          \[\leadsto \mathsf{fma}\left(0 - t, \mathsf{fma}\left(0.5, c\_p, \color{blue}{c\_n \cdot -0.5}\right), 1\right) \]

   8. Simplified 3.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0 - t, \mathsf{fma}\left(0.5, c\_p, c\_n \cdot -0.5\right), 1\right)} \]

   9. Taylor expanded in c_p around inf

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

       1. associate-*r* N/A

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

       2. metadata-eval N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. associate-*r* N/A

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

       7. metadata-eval N/A

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

       8. *-lowering-*.f64 73.1

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

   11. Simplified 73.1%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 5600:\\ \;\;\;\;\mathsf{fma}\left(c\_n \cdot t, 0.5, \mathsf{fma}\left(0 - t, 0.5 \cdot c\_p, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(c\_p \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.8% accurate, 42.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 5600:\\ \;\;\;\;\mathsf{fma}\left(t, -0.5 \cdot \left(c\_p - c\_n\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(c\_p \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= s 5600.0) (fma t (* -0.5 (- c_p c_n)) 1.0) (* t (* c_p -0.5))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (s <= 5600.0) {
		tmp = fma(t, (-0.5 * (c_p - c_n)), 1.0);
	} else {
		tmp = t * (c_p * -0.5);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (s <= 5600.0)
		tmp = fma(t, Float64(-0.5 * Float64(c_p - c_n)), 1.0);
	else
		tmp = Float64(t * Float64(c_p * -0.5));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[s, 5600.0], N[(t * N[(-0.5 * N[(c$95$p - c$95$n), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(t * N[(c$95$p * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < 5600

   1. Initial program 93.3%

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

   3. Taylor expanded in s around 0

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

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

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

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

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

      3. pow-lowering-pow.f64 94.1

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

   5. Simplified 94.1%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. neg-mul-1 N/A

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

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

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

      5. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - t}, \frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - t}, \frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(0 - t, \color{blue}{\frac{1}{2} \cdot c\_p + \frac{-1}{2} \cdot c\_n}, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(0 - t, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, c\_p, \frac{-1}{2} \cdot c\_n\right)}, 1\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(0 - t, \mathsf{fma}\left(\frac{1}{2}, c\_p, \color{blue}{c\_n \cdot \frac{-1}{2}}\right), 1\right) \]

      10. *-lowering-*.f64 97.2

          \[\leadsto \mathsf{fma}\left(0 - t, \mathsf{fma}\left(0.5, c\_p, \color{blue}{c\_n \cdot -0.5}\right), 1\right) \]

   8. Simplified 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0 - t, \mathsf{fma}\left(0.5, c\_p, c\_n \cdot -0.5\right), 1\right)} \]

   9. Taylor expanded in t around 0

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

       1. +-commutative N/A

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

       2. mul-1-neg N/A

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

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

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

       4. mul-1-neg N/A

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

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

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

       6. +-commutative N/A

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

       7. distribute-lft-in N/A

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

       8. associate-*r* N/A

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

       9. metadata-eval N/A

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

       10. associate-*r* N/A

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

       11. metadata-eval N/A

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 97.2

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

   11. Simplified 97.2%

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

   12. Taylor expanded in c_p around 0

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

       1. metadata-eval N/A

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

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

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

       3. distribute-lft-out-- N/A

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

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

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

       5. --lowering--.f64 97.2

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

   14. Simplified 97.2%

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

   if 5600 < s

   1. Initial program 0.0%

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

   3. Taylor expanded in s around 0

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

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

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

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

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

      3. pow-lowering-pow.f64 0.0

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

   5. Simplified 0.0%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. neg-mul-1 N/A

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

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

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

      5. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - t}, \frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - t}, \frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(0 - t, \color{blue}{\frac{1}{2} \cdot c\_p + \frac{-1}{2} \cdot c\_n}, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(0 - t, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, c\_p, \frac{-1}{2} \cdot c\_n\right)}, 1\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(0 - t, \mathsf{fma}\left(\frac{1}{2}, c\_p, \color{blue}{c\_n \cdot \frac{-1}{2}}\right), 1\right) \]

      10. *-lowering-*.f64 3.1

          \[\leadsto \mathsf{fma}\left(0 - t, \mathsf{fma}\left(0.5, c\_p, \color{blue}{c\_n \cdot -0.5}\right), 1\right) \]

   8. Simplified 3.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0 - t, \mathsf{fma}\left(0.5, c\_p, c\_n \cdot -0.5\right), 1\right)} \]

   9. Taylor expanded in c_p around inf

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

       1. associate-*r* N/A

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

       2. metadata-eval N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. associate-*r* N/A

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

       7. metadata-eval N/A

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

       8. *-lowering-*.f64 73.1

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

   11. Simplified 73.1%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 5600:\\ \;\;\;\;\mathsf{fma}\left(t, -0.5 \cdot \left(c\_p - c\_n\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(c\_p \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.7% accurate, 49.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(t, c\_p \cdot -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(c\_p \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= s 1000.0) (fma t (* c_p -0.5) 1.0) (* t (* c_p -0.5))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (s <= 1000.0) {
		tmp = fma(t, (c_p * -0.5), 1.0);
	} else {
		tmp = t * (c_p * -0.5);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (s <= 1000.0)
		tmp = fma(t, Float64(c_p * -0.5), 1.0);
	else
		tmp = Float64(t * Float64(c_p * -0.5));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[s, 1000.0], N[(t * N[(c$95$p * -0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(t * N[(c$95$p * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < 1e3

   1. Initial program 93.3%

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

   3. Taylor expanded in s around 0

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

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

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

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

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

      3. pow-lowering-pow.f64 94.1

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

   5. Simplified 94.1%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. neg-mul-1 N/A

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

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

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

      5. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - t}, \frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - t}, \frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(0 - t, \color{blue}{\frac{1}{2} \cdot c\_p + \frac{-1}{2} \cdot c\_n}, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(0 - t, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, c\_p, \frac{-1}{2} \cdot c\_n\right)}, 1\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(0 - t, \mathsf{fma}\left(\frac{1}{2}, c\_p, \color{blue}{c\_n \cdot \frac{-1}{2}}\right), 1\right) \]

      10. *-lowering-*.f64 97.2

          \[\leadsto \mathsf{fma}\left(0 - t, \mathsf{fma}\left(0.5, c\_p, \color{blue}{c\_n \cdot -0.5}\right), 1\right) \]

   8. Simplified 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0 - t, \mathsf{fma}\left(0.5, c\_p, c\_n \cdot -0.5\right), 1\right)} \]

   9. Taylor expanded in c_n around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. metadata-eval N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

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

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

       7. associate-*r* N/A

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

       8. metadata-eval N/A

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

       9. *-lowering-*.f64 97.1

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

   11. Simplified 97.1%

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

   if 1e3 < s

   1. Initial program 0.0%

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

   3. Taylor expanded in s around 0

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

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

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

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

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

      3. pow-lowering-pow.f64 0.0

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

   5. Simplified 0.0%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. neg-mul-1 N/A

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

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

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

      5. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - t}, \frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - t}, \frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(0 - t, \color{blue}{\frac{1}{2} \cdot c\_p + \frac{-1}{2} \cdot c\_n}, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(0 - t, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, c\_p, \frac{-1}{2} \cdot c\_n\right)}, 1\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(0 - t, \mathsf{fma}\left(\frac{1}{2}, c\_p, \color{blue}{c\_n \cdot \frac{-1}{2}}\right), 1\right) \]

      10. *-lowering-*.f64 3.1

          \[\leadsto \mathsf{fma}\left(0 - t, \mathsf{fma}\left(0.5, c\_p, \color{blue}{c\_n \cdot -0.5}\right), 1\right) \]

   8. Simplified 3.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0 - t, \mathsf{fma}\left(0.5, c\_p, c\_n \cdot -0.5\right), 1\right)} \]

   9. Taylor expanded in c_p around inf

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

       1. associate-*r* N/A

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

       2. metadata-eval N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. associate-*r* N/A

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

       7. metadata-eval N/A

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

       8. *-lowering-*.f64 73.1

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

   11. Simplified 73.1%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(t, c\_p \cdot -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(c\_p \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.8% accurate, 52.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 5600:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(c\_p \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= s 5600.0) 1.0 (* t (* c_p -0.5))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (s <= 5600.0) {
		tmp = 1.0;
	} else {
		tmp = t * (c_p * -0.5);
	}
	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 (s <= 5600.0d0) then
        tmp = 1.0d0
    else
        tmp = t * (c_p * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (s <= 5600.0) {
		tmp = 1.0;
	} else {
		tmp = t * (c_p * -0.5);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if s <= 5600.0:
		tmp = 1.0
	else:
		tmp = t * (c_p * -0.5)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (s <= 5600.0)
		tmp = 1.0;
	else
		tmp = Float64(t * Float64(c_p * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (s <= 5600.0)
		tmp = 1.0;
	else
		tmp = t * (c_p * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[s, 5600.0], 1.0, N[(t * N[(c$95$p * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < 5600

   1. Initial program 93.3%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. 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-sub0 N/A

          \[\leadsto \frac{{\left(1 + \frac{-1}{1 + e^{\color{blue}{0 - s}}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

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

          \[\leadsto \frac{{\left(1 + \frac{-1}{1 + e^{\color{blue}{0 - s}}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

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

          \[\leadsto \frac{{\left(1 + \frac{-1}{1 + e^{0 - s}}\right)}^{c\_n}}{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

   5. Simplified 96.0%

      \[\leadsto \color{blue}{\frac{{\left(1 + \frac{-1}{1 + e^{0 - s}}\right)}^{c\_n}}{{\left(1 + \frac{-1}{1 + e^{0 - t}}\right)}^{c\_n}}} \]

   6. Taylor expanded in c_n around 0

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

   8. Simplified 97.0%

      \[\leadsto \color{blue}{1} \]

   if 5600 < s

   1. Initial program 0.0%

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

   3. Taylor expanded in s around 0

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

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

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

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

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

      3. pow-lowering-pow.f64 0.0

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

   5. Simplified 0.0%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. neg-mul-1 N/A

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

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

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

      5. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - t}, \frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - t}, \frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(0 - t, \color{blue}{\frac{1}{2} \cdot c\_p + \frac{-1}{2} \cdot c\_n}, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(0 - t, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, c\_p, \frac{-1}{2} \cdot c\_n\right)}, 1\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(0 - t, \mathsf{fma}\left(\frac{1}{2}, c\_p, \color{blue}{c\_n \cdot \frac{-1}{2}}\right), 1\right) \]

      10. *-lowering-*.f64 3.1

          \[\leadsto \mathsf{fma}\left(0 - t, \mathsf{fma}\left(0.5, c\_p, \color{blue}{c\_n \cdot -0.5}\right), 1\right) \]

   8. Simplified 3.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0 - t, \mathsf{fma}\left(0.5, c\_p, c\_n \cdot -0.5\right), 1\right)} \]

   9. Taylor expanded in c_p around inf

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

       1. associate-*r* N/A

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

       2. metadata-eval N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. associate-*r* N/A

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

       7. metadata-eval N/A

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

       8. *-lowering-*.f64 73.1

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

   11. Simplified 73.1%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 5600:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(c\_p \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.2% 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 90.7%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
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-sub0 N/A

       \[\leadsto \frac{{\left(1 + \frac{-1}{1 + e^{\color{blue}{0 - s}}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

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

       \[\leadsto \frac{{\left(1 + \frac{-1}{1 + e^{\color{blue}{0 - s}}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

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

       \[\leadsto \frac{{\left(1 + \frac{-1}{1 + e^{0 - s}}\right)}^{c\_n}}{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

5. Simplified 93.8%

   \[\leadsto \color{blue}{\frac{{\left(1 + \frac{-1}{1 + e^{0 - s}}\right)}^{c\_n}}{{\left(1 + \frac{-1}{1 + e^{0 - t}}\right)}^{c\_n}}} \]

6. Taylor expanded in c_n around 0

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

8. Simplified 94.4%

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

Developer Target 1: 96.3% 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 2024199 
(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))))