Harley's example

Percentage Accurate: 91.6% → 98.9%

Time: 50.7s

Alternatives: 5

Speedup: 896.0×

• could not determine a ground truth

  1. c_p = 5.391391075751134e+82
  2. c_n = 2.5069360593886372e-33
  3. t = 7.5105740702111905e+53
  4. s = -6.713035979532155e+247

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: 91.6% 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: 98.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{-s}\\ t_2 := e^{-t}\\ t_3 := \mathsf{log1p}\left(\frac{-1}{1 + t\_2}\right)\\ \mathbf{if}\;c\_p \leq 5 \cdot 10^{-21}:\\ \;\;\;\;e^{c\_n \cdot \left(\mathsf{log1p}\left(\frac{-1}{1 + t\_1}\right) - t\_3\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(c\_p, \mathsf{log1p}\left(t\_2\right) - \mathsf{log1p}\left(t\_1\right), c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{-1 - t\_1}\right) - t\_3\right)\right)}\\ \end{array} \end{array} \]

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 (<= c_p 5e-21)
     (exp (* c_n (- (log1p (/ -1.0 (+ 1.0 t_1))) t_3)))
     (exp
      (fma
       c_p
       (- (log1p t_2) (log1p t_1))
       (* c_n (- (log1p (/ 1.0 (- -1.0 t_1))) t_3)))))))

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 (c_p <= 5e-21) {
		tmp = exp((c_n * (log1p((-1.0 / (1.0 + t_1))) - t_3)));
	} else {
		tmp = exp(fma(c_p, (log1p(t_2) - log1p(t_1)), (c_n * (log1p((1.0 / (-1.0 - t_1))) - t_3))));
	}
	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 (c_p <= 5e-21)
		tmp = exp(Float64(c_n * Float64(log1p(Float64(-1.0 / Float64(1.0 + t_1))) - t_3)));
	else
		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))));
	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[c$95$p, 5e-21], N[Exp[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], 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]]]]]

Derivation

1. Split input into 2 regimes

2. if c_p < 4.99999999999999973e-21

   1. Initial program 92.3%

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

   4. Applied rewrites 95.3%

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

   5. Taylor expanded in c_p around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. sub-neg N/A

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

      12. lower-log1p.f64 N/A

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

      13. distribute-neg-frac N/A

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

   7. Applied rewrites 98.7%

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

   if 4.99999999999999973e-21 < c_p

   1. Initial program 59.2%

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

   4. Applied rewrites 100.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_p \leq 5 \cdot 10^{-21}:\\ \;\;\;\;e^{c\_n \cdot \left(\mathsf{log1p}\left(\frac{-1}{1 + e^{-s}}\right) - \mathsf{log1p}\left(\frac{-1}{1 + e^{-t}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(c\_p, \mathsf{log1p}\left(e^{-t}\right) - \mathsf{log1p}\left(e^{-s}\right), c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{-1 - e^{-s}}\right) - \mathsf{log1p}\left(\frac{-1}{1 + e^{-t}}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 + e^{-t}\\ t_2 := 1 + e^{-s}\\ \mathbf{if}\;c\_p \leq 2 \cdot 10^{-27}:\\ \;\;\;\;e^{c\_n \cdot \left(\mathsf{log1p}\left(\frac{-1}{t\_2}\right) - \mathsf{log1p}\left(\frac{-1}{t\_1}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(t\_1 \cdot \frac{1}{t\_2}\right)}^{c\_p}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (+ 1.0 (exp (- t)))) (t_2 (+ 1.0 (exp (- s)))))
   (if (<= c_p 2e-27)
     (exp (* c_n (- (log1p (/ -1.0 t_2)) (log1p (/ -1.0 t_1)))))
     (pow (* t_1 (/ 1.0 t_2)) c_p))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = 1.0 + exp(-t);
	double t_2 = 1.0 + exp(-s);
	double tmp;
	if (c_p <= 2e-27) {
		tmp = exp((c_n * (log1p((-1.0 / t_2)) - log1p((-1.0 / t_1)))));
	} else {
		tmp = pow((t_1 * (1.0 / t_2)), c_p);
	}
	return tmp;
}

Java

public static double code(double c_p, double c_n, double t, double s) {
	double t_1 = 1.0 + Math.exp(-t);
	double t_2 = 1.0 + Math.exp(-s);
	double tmp;
	if (c_p <= 2e-27) {
		tmp = Math.exp((c_n * (Math.log1p((-1.0 / t_2)) - Math.log1p((-1.0 / t_1)))));
	} else {
		tmp = Math.pow((t_1 * (1.0 / t_2)), c_p);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	t_1 = 1.0 + math.exp(-t)
	t_2 = 1.0 + math.exp(-s)
	tmp = 0
	if c_p <= 2e-27:
		tmp = math.exp((c_n * (math.log1p((-1.0 / t_2)) - math.log1p((-1.0 / t_1)))))
	else:
		tmp = math.pow((t_1 * (1.0 / t_2)), c_p)
	return tmp

Julia

function code(c_p, c_n, t, s)
	t_1 = Float64(1.0 + exp(Float64(-t)))
	t_2 = Float64(1.0 + exp(Float64(-s)))
	tmp = 0.0
	if (c_p <= 2e-27)
		tmp = exp(Float64(c_n * Float64(log1p(Float64(-1.0 / t_2)) - log1p(Float64(-1.0 / t_1)))));
	else
		tmp = Float64(t_1 * Float64(1.0 / t_2)) ^ c_p;
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[(1.0 + N[Exp[(-t)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c$95$p, 2e-27], N[Exp[N[(c$95$n * N[(N[Log[1 + N[(-1.0 / t$95$2), $MachinePrecision]], $MachinePrecision] - N[Log[1 + N[(-1.0 / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(t$95$1 * N[(1.0 / t$95$2), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c_p < 2.0000000000000001e-27

   1. Initial program 94.0%

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

   4. Applied rewrites 96.5%

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

   5. Taylor expanded in c_p around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. sub-neg N/A

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

      12. lower-log1p.f64 N/A

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

      13. distribute-neg-frac N/A

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

   7. Applied rewrites 99.4%

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

   if 2.0000000000000001e-27 < c_p

   1. Initial program 76.1%

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

   3. Taylor expanded in c_n around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 78.7

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

   5. Applied rewrites 78.7%

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

   7. Applied rewrites 96.3%

      \[\leadsto {\left(\left(e^{-t} + 1\right) \cdot \frac{1}{e^{-s} + 1}\right)}^{\color{blue}{c\_p}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_p \leq 2 \cdot 10^{-27}:\\ \;\;\;\;e^{c\_n \cdot \left(\mathsf{log1p}\left(\frac{-1}{1 + e^{-s}}\right) - \mathsf{log1p}\left(\frac{-1}{1 + e^{-t}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(1 + e^{-t}\right) \cdot \frac{1}{1 + e^{-s}}\right)}^{c\_p}\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 96.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024223 
(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))))