Harley's example

Percentage Accurate: 91.0% → 96.7%

Time: 52.2s

Alternatives: 5

Speedup: 896.0×

• could not determine a ground truth

  1. c_p = 1.9450957683088803e-260
  2. c_n = 9.403407875407713e-125
  3. t = -2.5378109296081453e+25
  4. s = -1.581407546198402e-17

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.7% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.3%

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

3. Taylor expanded in c_n around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-neg.f64 92.3

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

5. Simplified 92.3%

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

7. Applied egg-rr 95.8%

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

8. Taylor expanded in t around 0

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

   1. neg-mul-1 N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 96.0

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

10. Simplified 96.0%

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

11. Taylor expanded in s around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. sub-neg N/A

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

    4. *-commutative N/A

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

    5. metadata-eval N/A

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

    6. lower-fma.f64 97.1

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

13. Simplified 97.1%

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

Alternative 2: 94.7% accurate, 32.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.3%

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

3. Taylor expanded in c_n around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-neg.f64 92.3

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

5. Simplified 92.3%

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

7. Applied egg-rr 95.8%

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

8. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. associate-*r* N/A

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

   7. distribute-lft1-in N/A

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

   8. lower-*.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. associate-*r/ N/A

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

   14. lower-pow.f64 N/A

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

10. Simplified 96.4%

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

11. Taylor expanded in s around 0

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

    1. associate-*r* N/A

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

    2. +-commutative N/A

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

    3. *-commutative N/A

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

    4. lower-fma.f64 N/A

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

    5. *-commutative N/A

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

    6. lower-*.f64 94.8

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

13. Simplified 94.8%

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

14. Final simplification 94.8%

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

Alternative 3: 94.6% accurate, 49.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.3%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
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. lower-/.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. lower-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. lower-+.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. lower-/.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. lower-+.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. lower-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. lower-neg.f64 N/A

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

   11. lower-pow.f64 N/A

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

5. Simplified 94.0%

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

6. Taylor expanded in c_n around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

8. Simplified 94.0%

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

9. Taylor expanded in s around 0

   \[\leadsto \mathsf{fma}\left(c\_n, \mathsf{log1p}\left(\color{blue}{\frac{-1}{2}}\right) - \mathsf{log1p}\left(\frac{-1}{1 + e^{\mathsf{neg}\left(t\right)}}\right), 1\right) \]
10. Step-by-step derivation

11. Simplified 94.4%

    \[\leadsto \mathsf{fma}\left(c\_n, \mathsf{log1p}\left(\color{blue}{-0.5}\right) - \mathsf{log1p}\left(\frac{-1}{1 + e^{-t}}\right), 1\right) \]

12. Taylor expanded in t around 0

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

    1. +-commutative N/A

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

    2. distribute-rgt-in N/A

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

    3. *-commutative N/A

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

    4. +-commutative N/A

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

    5. associate-*r* N/A

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

    6. associate-*r* N/A

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

    7. *-commutative N/A

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

    8. distribute-rgt-out N/A

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

    9. lower-fma.f64 N/A

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

    10. *-commutative N/A

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

    11. lower-*.f64 N/A

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

    12. +-commutative N/A

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

    13. *-commutative N/A

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

    14. lower-fma.f64 94.8

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

14. Simplified 94.8%

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

Alternative 4: 94.7% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.3%

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

3. Taylor expanded in c_n around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-neg.f64 92.3

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

5. Simplified 92.3%

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

7. Applied egg-rr 95.8%

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

8. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. associate-*r* N/A

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

   7. distribute-lft1-in N/A

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

   8. lower-*.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. associate-*r/ N/A

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

   14. lower-pow.f64 N/A

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

10. Simplified 96.4%

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

11. Taylor expanded in s around 0

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

    1. +-commutative N/A

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

    2. associate-*r* N/A

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

    3. *-commutative N/A

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

    4. associate-*r* N/A

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

    5. lower-fma.f64 N/A

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

    6. lower-*.f64 94.8

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

13. Simplified 94.8%

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

14. Final simplification 94.8%

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

Alternative 5: 94.6% 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 90.3%

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

3. Taylor expanded in c_n around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-neg.f64 92.3

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

5. Simplified 92.3%

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

6. Taylor expanded in c_p around 0

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

8. Simplified 94.8%

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

Developer Target 1: 96.7% 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 2024207 
(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))))