Harley's example

Percentage Accurate: 91.0% → 97.9%

Time: 52.8s

Alternatives: 10

Speedup: 896.0×

• could not determine a ground truth

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

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{-t}\\ t_2 := e^{-s}\\ \mathbf{if}\;-t \leq 4 \cdot 10^{-5}:\\ \;\;\;\;e^{\mathsf{fma}\left(c\_p, -\mathsf{log1p}\left(t\_2\right), c\_n \cdot \mathsf{log1p}\left(\frac{1}{-1 - t\_2}\right)\right) - \mathsf{fma}\left(c\_p, -\mathsf{log1p}\left(t\_1\right), c\_n \cdot \mathsf{log1p}\left(\frac{1}{-1 - t\_1}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(c\_n, t \cdot 0.5, s \cdot \left(c\_n \cdot -0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (exp (- t))) (t_2 (exp (- s))))
   (if (<= (- t) 4e-5)
     (exp
      (-
       (fma c_p (- (log1p t_2)) (* c_n (log1p (/ 1.0 (- -1.0 t_2)))))
       (fma c_p (- (log1p t_1)) (* c_n (log1p (/ 1.0 (- -1.0 t_1)))))))
     (exp (fma c_n (* t 0.5) (* s (* c_n -0.5)))))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = exp(-t);
	double t_2 = exp(-s);
	double tmp;
	if (-t <= 4e-5) {
		tmp = exp((fma(c_p, -log1p(t_2), (c_n * log1p((1.0 / (-1.0 - t_2))))) - fma(c_p, -log1p(t_1), (c_n * log1p((1.0 / (-1.0 - t_1)))))));
	} else {
		tmp = exp(fma(c_n, (t * 0.5), (s * (c_n * -0.5))));
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	t_1 = exp(Float64(-t))
	t_2 = exp(Float64(-s))
	tmp = 0.0
	if (Float64(-t) <= 4e-5)
		tmp = exp(Float64(fma(c_p, Float64(-log1p(t_2)), Float64(c_n * log1p(Float64(1.0 / Float64(-1.0 - t_2))))) - fma(c_p, Float64(-log1p(t_1)), Float64(c_n * log1p(Float64(1.0 / Float64(-1.0 - t_1)))))));
	else
		tmp = exp(fma(c_n, Float64(t * 0.5), Float64(s * Float64(c_n * -0.5))));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[Exp[(-t)], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-s)], $MachinePrecision]}, If[LessEqual[(-t), 4e-5], N[Exp[N[(N[(c$95$p * (-N[Log[1 + t$95$2], $MachinePrecision]) + N[(c$95$n * N[Log[1 + N[(1.0 / N[(-1.0 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c$95$p * (-N[Log[1 + t$95$1], $MachinePrecision]) + N[(c$95$n * N[Log[1 + N[(1.0 / N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(c$95$n * N[(t * 0.5), $MachinePrecision] + N[(s * N[(c$95$n * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 t) < 4.00000000000000033e-5

   1. Initial program 93.1%

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

   4. Applied rewrites 99.9%

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

   if 4.00000000000000033e-5 < (neg.f64 t)

   1. Initial program 45.5%

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in c_p around 0

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

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-log1p.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. sub-neg N/A

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

      13. lower-log1p.f64 N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in t around 0

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

   13. Applied rewrites 100.0%

       \[\leadsto e^{\mathsf{fma}\left(c\_n, 0.5 \cdot t, s \cdot \left(c\_n \cdot -0.5\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-t \leq 4 \cdot 10^{-5}:\\ \;\;\;\;e^{\mathsf{fma}\left(c\_p, -\mathsf{log1p}\left(e^{-s}\right), c\_n \cdot \mathsf{log1p}\left(\frac{1}{-1 - e^{-s}}\right)\right) - \mathsf{fma}\left(c\_p, -\mathsf{log1p}\left(e^{-t}\right), c\_n \cdot \mathsf{log1p}\left(\frac{1}{-1 - e^{-t}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(c\_n, t \cdot 0.5, s \cdot \left(c\_n \cdot -0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{-s}\\ t_2 := e^{-t}\\ t_3 := \mathsf{log1p}\left(\frac{1}{-1 - t\_2}\right)\\ \mathbf{if}\;-t \leq 4 \cdot 10^{-163}:\\ \;\;\;\;e^{\mathsf{fma}\left(c\_p, \mathsf{log1p}\left(t\_2\right) - \mathsf{log1p}\left(t\_1\right), c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{-1 - t\_1}\right) - t\_3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(c\_n, \log 0.5 - t\_3, s \cdot \left(c\_n \cdot -0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (exp (- s))) (t_2 (exp (- t))) (t_3 (log1p (/ 1.0 (- -1.0 t_2)))))
   (if (<= (- t) 4e-163)
     (exp
      (fma
       c_p
       (- (log1p t_2) (log1p t_1))
       (* c_n (- (log1p (/ 1.0 (- -1.0 t_1))) t_3))))
     (exp (fma c_n (- (log 0.5) t_3) (* s (* c_n -0.5)))))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = exp(-s);
	double t_2 = exp(-t);
	double t_3 = log1p((1.0 / (-1.0 - t_2)));
	double tmp;
	if (-t <= 4e-163) {
		tmp = exp(fma(c_p, (log1p(t_2) - log1p(t_1)), (c_n * (log1p((1.0 / (-1.0 - t_1))) - t_3))));
	} else {
		tmp = exp(fma(c_n, (log(0.5) - t_3), (s * (c_n * -0.5))));
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	t_1 = exp(Float64(-s))
	t_2 = exp(Float64(-t))
	t_3 = log1p(Float64(1.0 / Float64(-1.0 - t_2)))
	tmp = 0.0
	if (Float64(-t) <= 4e-163)
		tmp = exp(fma(c_p, Float64(log1p(t_2) - log1p(t_1)), Float64(c_n * Float64(log1p(Float64(1.0 / Float64(-1.0 - t_1))) - t_3))));
	else
		tmp = exp(fma(c_n, Float64(log(0.5) - t_3), Float64(s * Float64(c_n * -0.5))));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[Exp[(-s)], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-t)], $MachinePrecision]}, Block[{t$95$3 = N[Log[1 + N[(1.0 / N[(-1.0 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[(-t), 4e-163], N[Exp[N[(c$95$p * N[(N[Log[1 + t$95$2], $MachinePrecision] - N[Log[1 + t$95$1], $MachinePrecision]), $MachinePrecision] + N[(c$95$n * N[(N[Log[1 + N[(1.0 / N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(c$95$n * N[(N[Log[0.5], $MachinePrecision] - t$95$3), $MachinePrecision] + N[(s * N[(c$95$n * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 92.7%

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

   4. Applied rewrites 97.9%

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

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

   1. Initial program 86.9%

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

   4. Applied rewrites 91.4%

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

   5. Taylor expanded in c_p around 0

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

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-log1p.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. sub-neg N/A

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

      13. lower-log1p.f64 N/A

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

   7. Applied rewrites 96.7%

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

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 97.0%

       \[\leadsto e^{\mathsf{fma}\left(c\_n, \color{blue}{\log 0.5 - \mathsf{log1p}\left(\frac{-1}{1 + e^{-t}}\right)}, s \cdot \left(c\_n \cdot -0.5\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-t \leq 4 \cdot 10^{-163}:\\ \;\;\;\;e^{\mathsf{fma}\left(c\_p, \mathsf{log1p}\left(e^{-t}\right) - \mathsf{log1p}\left(e^{-s}\right), c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{-1 - e^{-s}}\right) - \mathsf{log1p}\left(\frac{1}{-1 - e^{-t}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(c\_n, \log 0.5 - \mathsf{log1p}\left(\frac{1}{-1 - e^{-t}}\right), s \cdot \left(c\_n \cdot -0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 96.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024219 
(FPCore (c_p c_n t s)
  :name "Harley's example"
  :precision binary64
  :pre (and (< 0.0 c_p) (< 0.0 c_n))

  :alt
  (! :herbie-platform default (* (pow (/ (+ 1 (exp (- t))) (+ 1 (exp (- s)))) c_p) (pow (/ (+ 1 (exp t)) (+ 1 (exp s))) c_n)))

  (/ (* (pow (/ 1.0 (+ 1.0 (exp (- s)))) c_p) (pow (- 1.0 (/ 1.0 (+ 1.0 (exp (- s))))) c_n)) (* (pow (/ 1.0 (+ 1.0 (exp (- t)))) c_p) (pow (- 1.0 (/ 1.0 (+ 1.0 (exp (- t))))) c_n))))