Harley's example

Percentage Accurate: 90.3% → 96.6%

Time: 41.9s

Alternatives: 6

Speedup: 896.0×

• could not determine a ground truth

  1. c_p = 2.7098115727851685e-272
  2. c_n = 1.3571497222431857e-218
  3. t = -1.0413163700722758e+68
  4. s = -4.461370179000461e+207

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: 90.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ e^{\left(\mathsf{log1p}\left(\frac{-1}{e^{-s} + 1}\right) - \mathsf{log1p}\left(\frac{-1}{e^{-t} - -1}\right)\right) \cdot c\_n} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (exp
  (*
   (-
    (log1p (/ -1.0 (+ (exp (- s)) 1.0)))
    (log1p (/ -1.0 (- (exp (- t)) -1.0))))
   c_n)))

C

double code(double c_p, double c_n, double t, double s) {
	return exp(((log1p((-1.0 / (exp(-s) + 1.0))) - log1p((-1.0 / (exp(-t) - -1.0)))) * c_n));
}

Java

public static double code(double c_p, double c_n, double t, double s) {
	return Math.exp(((Math.log1p((-1.0 / (Math.exp(-s) + 1.0))) - Math.log1p((-1.0 / (Math.exp(-t) - -1.0)))) * c_n));
}

Python

def code(c_p, c_n, t, s):
	return math.exp(((math.log1p((-1.0 / (math.exp(-s) + 1.0))) - math.log1p((-1.0 / (math.exp(-t) - -1.0)))) * c_n))

Julia

function code(c_p, c_n, t, s)
	return exp(Float64(Float64(log1p(Float64(-1.0 / Float64(exp(Float64(-s)) + 1.0))) - log1p(Float64(-1.0 / Float64(exp(Float64(-t)) - -1.0)))) * c_n))
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[Exp[N[(N[(N[Log[1 + N[(-1.0 / N[(N[Exp[(-s)], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[1 + N[(-1.0 / N[(N[Exp[(-t)], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c$95$n), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 91.5%

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

4. Applied rewrites 96.2%

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

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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

7. Applied rewrites 98.9%

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

8. Final simplification 98.9%

   \[\leadsto e^{\left(\mathsf{log1p}\left(\frac{-1}{e^{-s} + 1}\right) - \mathsf{log1p}\left(\frac{-1}{e^{-t} - -1}\right)\right) \cdot c\_n} \]
9. Add Preprocessing

Alternative 2: 95.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c\_n \leq 740000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, t, 0.5\right) \cdot c\_n, t, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(1 - \frac{-1}{e^{-t} - -1}\right)}^{c\_n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= c_n 740000.0)
   (fma (* (fma 0.125 t 0.5) c_n) t 1.0)
   (/ (pow 0.5 c_n) (pow (- 1.0 (/ -1.0 (- (exp (- t)) -1.0))) c_n))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_n <= 740000.0) {
		tmp = fma((fma(0.125, t, 0.5) * c_n), t, 1.0);
	} else {
		tmp = pow(0.5, c_n) / pow((1.0 - (-1.0 / (exp(-t) - -1.0))), c_n);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (c_n <= 740000.0)
		tmp = fma(Float64(fma(0.125, t, 0.5) * c_n), t, 1.0);
	else
		tmp = Float64((0.5 ^ c_n) / (Float64(1.0 - Float64(-1.0 / Float64(exp(Float64(-t)) - -1.0))) ^ c_n));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[c$95$n, 740000.0], N[(N[(N[(0.125 * t + 0.5), $MachinePrecision] * c$95$n), $MachinePrecision] * t + 1.0), $MachinePrecision], N[(N[Power[0.5, c$95$n], $MachinePrecision] / N[Power[N[(1.0 - N[(-1.0 / N[(N[Exp[(-t)], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c_n < 7.4e5

   1. Initial program 94.8%

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

   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. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. neg-mul-1 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-pow.f64 N/A

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 97.2%

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

   9. Taylor expanded in t around 0

      \[\leadsto 1 + t \cdot \color{blue}{\left(-1 \cdot \left(t \cdot \left(\frac{-1}{4} \cdot {c\_n}^{2} + \left(\frac{-1}{8} \cdot c\_n + \frac{1}{8} \cdot {c\_n}^{2}\right)\right)\right) - \frac{-1}{2} \cdot c\_n\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 98.8%

       \[\leadsto \mathsf{fma}\left(-\mathsf{fma}\left(\mathsf{fma}\left(c\_n \cdot c\_n, -0.25, \mathsf{fma}\left(c\_n \cdot c\_n, 0.125, -0.125 \cdot c\_n\right)\right), t, -0.5 \cdot c\_n\right), t, 1\right) \]

   12. Taylor expanded in c_n around 0

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

   14. Applied rewrites 98.8%

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

   if 7.4e5 < c_n

   1. Initial program 0.0%

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

   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. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. neg-mul-1 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-pow.f64 N/A

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

   5. Applied rewrites 44.4%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 44.4%

      \[\leadsto \frac{{0.5}^{c\_n}}{\color{blue}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c\_n}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 78.5%

       \[\leadsto \frac{{0.5}^{c\_n}}{{\left(1 - \frac{1}{\left(e^{-t} + 1\right) \cdot -1}\right)}^{c\_n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_n \leq 740000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, t, 0.5\right) \cdot c\_n, t, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(1 - \frac{-1}{e^{-t} - -1}\right)}^{c\_n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c\_n \leq 0.55:\\ \;\;\;\;\frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c\_n}}{{0.5}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-\mathsf{log1p}\left(e^{-t}\right), c\_p, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= c_n 0.55)
   (/ (pow (- 1.0 (/ 1.0 (+ (exp (- s)) 1.0))) c_n) (pow 0.5 c_n))
   (/ 1.0 (fma (- (log1p (exp (- t)))) c_p 1.0))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_n <= 0.55) {
		tmp = pow((1.0 - (1.0 / (exp(-s) + 1.0))), c_n) / pow(0.5, c_n);
	} else {
		tmp = 1.0 / fma(-log1p(exp(-t)), c_p, 1.0);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (c_n <= 0.55)
		tmp = Float64((Float64(1.0 - Float64(1.0 / Float64(exp(Float64(-s)) + 1.0))) ^ c_n) / (0.5 ^ c_n));
	else
		tmp = Float64(1.0 / fma(Float64(-log1p(exp(Float64(-t)))), c_p, 1.0));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[c$95$n, 0.55], N[(N[Power[N[(1.0 - N[(1.0 / N[(N[Exp[(-s)], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$n], $MachinePrecision] / N[Power[0.5, c$95$n], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[((-N[Log[1 + N[Exp[(-t)], $MachinePrecision]], $MachinePrecision]) * c$95$p + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c_n < 0.55000000000000004

   1. Initial program 95.9%

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

   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. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. neg-mul-1 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-pow.f64 N/A

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in t around 0

      \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c\_n}}{{\frac{1}{2}}^{c\_n}} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.1%

      \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c\_n}}{{0.5}^{c\_n}} \]

   if 0.55000000000000004 < c_n

   1. Initial program 25.0%

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

   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. neg-mul-1 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 57.2

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

   5. Applied rewrites 57.2%

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

   6. Taylor expanded in c_p around 0

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

   8. Applied rewrites 57.2%

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

   9. Taylor expanded in c_p around 0

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

   11. Applied rewrites 81.8%

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

Alternative 4: 94.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ e^{\mathsf{fma}\left(\log 0.5, c\_n, \left(-c\_n\right) \cdot \log \left(\mathsf{fma}\left(t, -0.25, 0.5\right)\right)\right)} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (exp (fma (log 0.5) c_n (* (- c_n) (log (fma t -0.25 0.5))))))

C

double code(double c_p, double c_n, double t, double s) {
	return exp(fma(log(0.5), c_n, (-c_n * log(fma(t, -0.25, 0.5)))));
}

Julia

function code(c_p, c_n, t, s)
	return exp(fma(log(0.5), c_n, Float64(Float64(-c_n) * log(fma(t, -0.25, 0.5)))))
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[Exp[N[(N[Log[0.5], $MachinePrecision] * c$95$n + N[((-c$95$n) * N[Log[N[(t * -0.25 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 91.5%

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

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. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. neg-mul-1 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. neg-mul-1 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-pow.f64 N/A

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

5. Applied rewrites 95.7%

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

6. Taylor expanded in s around 0

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

8. Applied rewrites 95.4%

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

9. Taylor expanded in t around 0

   \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(\frac{1}{2} + \frac{-1}{4} \cdot t\right)}^{c\_n}} \]
10. Step-by-step derivation

11. Applied rewrites 94.9%

    \[\leadsto \frac{{0.5}^{c\_n}}{{\left(\mathsf{fma}\left(-0.25, t, 0.5\right)\right)}^{c\_n}} \]
12. Step-by-step derivation

13. Applied rewrites 96.2%

    \[\leadsto e^{\mathsf{fma}\left(\log 0.5, c\_n, -\log \left(\mathsf{fma}\left(t, -0.25, 0.5\right)\right) \cdot c\_n\right)} \]

14. Final simplification 96.2%

    \[\leadsto e^{\mathsf{fma}\left(\log 0.5, c\_n, \left(-c\_n\right) \cdot \log \left(\mathsf{fma}\left(t, -0.25, 0.5\right)\right)\right)} \]
15. Add Preprocessing

Alternative 5: 94.0% accurate, 49.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.5%

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

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. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. neg-mul-1 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. neg-mul-1 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-pow.f64 N/A

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

5. Applied rewrites 95.7%

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

6. Taylor expanded in s around 0

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

8. Applied rewrites 95.4%

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

9. Taylor expanded in t around 0

   \[\leadsto 1 + t \cdot \color{blue}{\left(-1 \cdot \left(t \cdot \left(\frac{-1}{4} \cdot {c\_n}^{2} + \left(\frac{-1}{8} \cdot c\_n + \frac{1}{8} \cdot {c\_n}^{2}\right)\right)\right) - \frac{-1}{2} \cdot c\_n\right)} \]
10. Step-by-step derivation

11. Applied rewrites 96.1%

    \[\leadsto \mathsf{fma}\left(-\mathsf{fma}\left(\mathsf{fma}\left(c\_n \cdot c\_n, -0.25, \mathsf{fma}\left(c\_n \cdot c\_n, 0.125, -0.125 \cdot c\_n\right)\right), t, -0.5 \cdot c\_n\right), t, 1\right) \]

12. Taylor expanded in c_n around 0

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

14. Applied rewrites 96.1%

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

Alternative 6: 94.1% 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 91.5%

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

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. neg-mul-1 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-exp.f64 N/A

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

   8. neg-mul-1 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-exp.f64 N/A

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

   15. lower-neg.f64 93.4

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

5. Applied rewrites 93.4%

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

6. Taylor expanded in c_p around 0

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

8. Applied rewrites 90.5%

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

9. Taylor expanded in c_p around 0

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

11. Applied rewrites 96.1%

    \[\leadsto 1 \]
12. 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 2024304 
(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))))