Harley's example

Percentage Accurate: 91.2% → 98.7%

Time: 53.0s

Alternatives: 3

Speedup: 896.0×

• could not determine a ground truth

  1. c_p = 1.544250578868645e-115
  2. c_n = 22939400067581120.0
  3. t = -1.6196907912047204e-168
  4. s = -1.554598615077534e-273

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: 91.2% 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: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{log1p}\left(e^{-t}\right)\\ \mathbf{if}\;c\_p \leq 5 \cdot 10^{-98}:\\ \;\;\;\;e^{\left(\left(-c\_n\right) \cdot \mathsf{fma}\left(\frac{\log \left(1 - e^{-\mathsf{log1p}\left(e^{-s}\right)}\right)}{c\_p}, -1, \frac{\log \left(1 - e^{-t\_1}\right)}{c\_p}\right)\right) \cdot c\_p}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-c\_p\right) \cdot \left(\mathsf{fma}\left(0.125 \cdot s - 0.5, s, \log 2\right) - t\_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (log1p (exp (- t)))))
   (if (<= c_p 5e-98)
     (exp
      (*
       (*
        (- c_n)
        (fma
         (/ (log (- 1.0 (exp (- (log1p (exp (- s))))))) c_p)
         -1.0
         (/ (log (- 1.0 (exp (- t_1)))) c_p)))
       c_p))
     (exp (* (- c_p) (- (fma (- (* 0.125 s) 0.5) s (log 2.0)) t_1))))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = log1p(exp(-t));
	double tmp;
	if (c_p <= 5e-98) {
		tmp = exp(((-c_n * fma((log((1.0 - exp(-log1p(exp(-s))))) / c_p), -1.0, (log((1.0 - exp(-t_1))) / c_p))) * c_p));
	} else {
		tmp = exp((-c_p * (fma(((0.125 * s) - 0.5), s, log(2.0)) - t_1)));
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	t_1 = log1p(exp(Float64(-t)))
	tmp = 0.0
	if (c_p <= 5e-98)
		tmp = exp(Float64(Float64(Float64(-c_n) * fma(Float64(log(Float64(1.0 - exp(Float64(-log1p(exp(Float64(-s))))))) / c_p), -1.0, Float64(log(Float64(1.0 - exp(Float64(-t_1)))) / c_p))) * c_p));
	else
		tmp = exp(Float64(Float64(-c_p) * Float64(fma(Float64(Float64(0.125 * s) - 0.5), s, log(2.0)) - t_1)));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[Log[1 + N[Exp[(-t)], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[c$95$p, 5e-98], N[Exp[N[(N[((-c$95$n) * N[(N[(N[Log[N[(1.0 - N[Exp[(-N[Log[1 + N[Exp[(-s)], $MachinePrecision]], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / c$95$p), $MachinePrecision] * -1.0 + N[(N[Log[N[(1.0 - N[Exp[(-t$95$1)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / c$95$p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c$95$p), $MachinePrecision]], $MachinePrecision], N[Exp[N[((-c$95$p) * N[(N[(N[(N[(0.125 * s), $MachinePrecision] - 0.5), $MachinePrecision] * s + N[Log[2.0], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c_p < 5.00000000000000018e-98

   1. Initial program 93.8%

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

   4. Applied rewrites 96.5%

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

   5. Taylor expanded in c_p around inf

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

   7. Applied rewrites 96.5%

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

   8. Taylor expanded in c_n around -inf

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

   10. Applied rewrites 99.9%

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

   if 5.00000000000000018e-98 < c_p

   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 97.6%

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

   5. Taylor expanded in c_p around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-log1p.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-log1p.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 98.8

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

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in s around 0

      \[\leadsto e^{\left(-c\_p\right) \cdot \left(\left(\log 2 + s \cdot \left(\frac{1}{8} \cdot s - \frac{1}{2}\right)\right) - \mathsf{log1p}\left(\color{blue}{e^{-t}}\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

       \[\leadsto e^{\left(-c\_p\right) \cdot \left(\mathsf{fma}\left(0.125 \cdot s - 0.5, s, \log 2\right) - \mathsf{log1p}\left(\color{blue}{e^{-t}}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ e^{\left(-c\_p\right) \cdot \left(\mathsf{fma}\left(0.125 \cdot s - 0.5, s, \log 2\right) - \mathsf{log1p}\left(e^{-t}\right)\right)} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (exp
  (* (- c_p) (- (fma (- (* 0.125 s) 0.5) s (log 2.0)) (log1p (exp (- t)))))))

C

double code(double c_p, double c_n, double t, double s) {
	return exp((-c_p * (fma(((0.125 * s) - 0.5), s, log(2.0)) - log1p(exp(-t)))));
}

Julia

function code(c_p, c_n, t, s)
	return exp(Float64(Float64(-c_p) * Float64(fma(Float64(Float64(0.125 * s) - 0.5), s, log(2.0)) - log1p(exp(Float64(-t))))))
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[Exp[N[((-c$95$p) * N[(N[(N[(N[(0.125 * s), $MachinePrecision] - 0.5), $MachinePrecision] * s + N[Log[2.0], $MachinePrecision]), $MachinePrecision] - N[Log[1 + N[Exp[(-t)], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 93.4%

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

4. Applied rewrites 96.9%

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

5. Taylor expanded in c_p around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-lft-neg-in N/A

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-log1p.f64 N/A

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

   7. lower-exp.f64 N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-log1p.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-neg.f64 96.8

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

7. Applied rewrites 96.8%

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

8. Taylor expanded in s around 0

   \[\leadsto e^{\left(-c\_p\right) \cdot \left(\left(\log 2 + s \cdot \left(\frac{1}{8} \cdot s - \frac{1}{2}\right)\right) - \mathsf{log1p}\left(\color{blue}{e^{-t}}\right)\right)} \]
9. Step-by-step derivation

10. Applied rewrites 98.0%

    \[\leadsto e^{\left(-c\_p\right) \cdot \left(\mathsf{fma}\left(0.125 \cdot s - 0.5, s, \log 2\right) - \mathsf{log1p}\left(\color{blue}{e^{-t}}\right)\right)} \]
11. Add Preprocessing

Alternative 3: 94.5% 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 93.4%

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

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

   5. lower-+.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-neg.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lower-exp.f64 N/A

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

   13. lower-neg.f64 95.7

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

5. Applied rewrites 95.7%

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

6. Taylor expanded in c_p around 0

   \[\leadsto 1 \]
7. Step-by-step derivation

8. Applied rewrites 96.4%

   \[\leadsto 1 \]
9. Add Preprocessing

Developer Target 1: 96.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024340 
(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))))