Harley's example

Percentage Accurate: 90.5% → 99.5%

Time: 53.2s

Alternatives: 9

Speedup: 896.0×

• could not determine a ground truth

  1. c_p = 1.969439565131657e+85
  2. c_n = 3.5239944257471017e-255
  3. t = 6.934768818192517e+104
  4. s = -2.6005553924347215e-137

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: 90.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.6%

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

4. Applied rewrites 94.1%

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

5. Taylor expanded in t around 0

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

   1. lower-fma.f64 N/A

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

   2. lower--.f64 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f64 N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-log.f64 N/A

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

7. Applied rewrites 98.1%

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

8. Taylor expanded in s around 0

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

   1. lower-fma.f64 N/A

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

10. Applied rewrites 99.5%

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

11. Final simplification 99.5%

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

Alternative 2: 99.4% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.6%

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

4. Applied rewrites 94.1%

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

5. Taylor expanded in t around 0

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

   1. lower-fma.f64 N/A

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

   2. lower--.f64 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f64 N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-log.f64 N/A

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

7. Applied rewrites 98.1%

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

8. Taylor expanded in s around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

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

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

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

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. associate-*r* N/A

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

   13. metadata-eval N/A

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

   14. associate-*r* N/A

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

   15. distribute-lft-in N/A

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

   16. metadata-eval N/A

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

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

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

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

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

   19. associate-*r* N/A

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

   20. metadata-eval N/A

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

   21. lower-*.f64 N/A

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

   22. lower--.f64 99.4

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

10. Applied rewrites 99.4%

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

Alternative 3: 98.7% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.6%

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

4. Applied rewrites 94.1%

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

5. Taylor expanded in t around 0

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

   1. lower-fma.f64 N/A

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

   2. lower--.f64 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f64 N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-log.f64 N/A

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

7. Applied rewrites 98.1%

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

8. Taylor expanded in s around 0

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

   1. lower-fma.f64 N/A

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

10. Applied rewrites 99.5%

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

11. Taylor expanded in c_n around inf

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

    1. lower-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. lower-*.f64 N/A

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

    5. sub-neg N/A

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

    6. *-commutative N/A

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

    7. metadata-eval N/A

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

    8. lower-fma.f64 98.6

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

13. Applied rewrites 98.6%

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

Alternative 4: 98.4% accurate, 7.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    code = exp(((-0.5d0) * (s * (c_n - c_p))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.6%

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

4. Applied rewrites 94.1%

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

5. Taylor expanded in t around 0

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

   1. lower-fma.f64 N/A

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

   2. lower--.f64 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f64 N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-log.f64 N/A

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

7. Applied rewrites 98.1%

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

8. Taylor expanded in s around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

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

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

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

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. associate-*r* N/A

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

   13. metadata-eval N/A

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

   14. associate-*r* N/A

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

   15. distribute-lft-in N/A

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

   16. metadata-eval N/A

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

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

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

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

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

   19. associate-*r* N/A

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

   20. metadata-eval N/A

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

   21. lower-*.f64 N/A

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

   22. lower--.f64 99.4

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

10. Applied rewrites 99.4%

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

11. Taylor expanded in t around 0

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

    1. lower-*.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. lower--.f64 98.1

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

13. Applied rewrites 98.1%

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

Alternative 5: 95.5% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    code = exp((0.5d0 * (c_n * t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.6%

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

4. Applied rewrites 94.1%

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

5. Taylor expanded in t around 0

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

   1. lower-fma.f64 N/A

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

   2. lower--.f64 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f64 N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-log.f64 N/A

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

7. Applied rewrites 98.1%

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

8. Taylor expanded in s around 0

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

   1. lower-exp.f64 N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

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

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

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

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 96.8

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

10. Applied rewrites 96.8%

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

11. Taylor expanded in c_p around 0

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

    1. lower-exp.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. *-commutative N/A

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

    4. lower-*.f64 96.3

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

13. Applied rewrites 96.3%

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

14. Final simplification 96.3%

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

Alternative 6: 94.3% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    code = exp(((-0.5d0) * (c_p * t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.6%

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

4. Applied rewrites 94.1%

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

5. Taylor expanded in t around 0

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

   1. lower-fma.f64 N/A

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

   2. lower--.f64 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f64 N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-log.f64 N/A

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

7. Applied rewrites 98.1%

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

8. Taylor expanded in s around 0

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

   1. lower-exp.f64 N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

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

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

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

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 96.8

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

10. Applied rewrites 96.8%

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

11. Taylor expanded in c_n around 0

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

    1. lower-exp.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. *-commutative N/A

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

    4. lower-*.f64 95.3

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

13. Applied rewrites 95.3%

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

14. Final simplification 95.3%

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

Alternative 7: 94.0% accurate, 24.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.6%

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

4. Applied rewrites 94.1%

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

5. Taylor expanded in t around 0

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

   1. lower-fma.f64 N/A

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

   2. lower--.f64 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f64 N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-log.f64 N/A

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

7. Applied rewrites 98.1%

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

8. Taylor expanded in s around 0

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

   1. lower-exp.f64 N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

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

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

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

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 96.8

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

10. Applied rewrites 96.8%

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

11. Taylor expanded in t around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. +-commutative N/A

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

    4. associate-*r* N/A

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

    5. lower-fma.f64 N/A

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

    6. lower-*.f64 N/A

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

    7. unpow2 N/A

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

    8. lower-*.f64 N/A

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

    9. lower--.f64 N/A

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

    10. lower--.f64 N/A

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

    11. lower-*.f64 N/A

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

    12. lower--.f64 94.9

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

13. Applied rewrites 94.9%

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

14. Final simplification 94.9%

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

Alternative 8: 94.0% accurate, 59.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.6%

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

4. Applied rewrites 94.1%

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

5. Taylor expanded in t around 0

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

   1. lower-fma.f64 N/A

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

   2. lower--.f64 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f64 N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-log.f64 N/A

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

7. Applied rewrites 98.1%

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

8. Taylor expanded in s around 0

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

   1. lower-exp.f64 N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

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

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

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

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 96.8

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

10. Applied rewrites 96.8%

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

11. Taylor expanded in t around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower--.f64 94.9

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

13. Applied rewrites 94.9%

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

14. Final simplification 94.9%

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

Alternative 9: 94.0% 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 89.6%

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

3. Taylor expanded in c_p around 0

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

   1. 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. Applied rewrites 95.6%

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

6. Taylor expanded in c_n around 0

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

8. Applied rewrites 94.8%

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

Developer Target 1: 96.0% 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 2024216 
(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))))