Harley's example

Percentage Accurate: 91.3% → 99.3%

Time: 51.8s

Alternatives: 9

Speedup: 896.0×

• could not determine a ground truth

  1. c_p = 6.732690877649198e+69
  2. c_n = 6.217766726422771e-101
  3. t = 5.363778019898235e-90
  4. s = 3.908712896624253e+62

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: 91.3% 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: 99.3% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq 2 \cdot 10^{-14}:\\ \;\;\;\;e^{\mathsf{fma}\left(-0.5, s, 0.5 \cdot t\right) \cdot c\_n}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(0.5 \cdot c\_p\right) \cdot s}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- s) 2e-14)
   (exp (* (fma -0.5 s (* 0.5 t)) c_n))
   (exp (* (* 0.5 c_p) s))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 2e-14) {
		tmp = exp((fma(-0.5, s, (0.5 * t)) * c_n));
	} else {
		tmp = exp(((0.5 * c_p) * s));
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (Float64(-s) <= 2e-14)
		tmp = exp(Float64(fma(-0.5, s, Float64(0.5 * t)) * c_n));
	else
		tmp = exp(Float64(Float64(0.5 * c_p) * s));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-s), 2e-14], N[Exp[N[(N[(-0.5 * s + N[(0.5 * t), $MachinePrecision]), $MachinePrecision] * c$95$n), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(0.5 * c$95$p), $MachinePrecision] * s), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 s) < 2e-14

   1. Initial program 91.8%

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in s around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in c_n around inf

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

   13. Applied rewrites 100.0%

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

   if 2e-14 < (neg.f64 s)

   1. Initial program 31.3%

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

   4. Applied rewrites 77.4%

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

   5. Taylor expanded in s around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 92.4%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 92.4%

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

   11. Taylor expanded in c_p around inf

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

   13. Applied rewrites 100.0%

       \[\leadsto e^{\left(0.5 \cdot c\_p\right) \cdot s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.6% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.7%

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

4. Applied rewrites 97.0%

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

5. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 99.6%

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

Alternative 3: 98.7% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq 10^{-15}:\\ \;\;\;\;e^{\left(-0.5 \cdot c\_n\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(0.5 \cdot c\_p\right) \cdot s}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- s) 1e-15) (exp (* (* -0.5 c_n) s)) (exp (* (* 0.5 c_p) s))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 1e-15) {
		tmp = exp(((-0.5 * c_n) * s));
	} else {
		tmp = exp(((0.5 * c_p) * s));
	}
	return tmp;
}

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) :: tmp
    if (-s <= 1d-15) then
        tmp = exp((((-0.5d0) * c_n) * s))
    else
        tmp = exp(((0.5d0 * c_p) * s))
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 1e-15) {
		tmp = Math.exp(((-0.5 * c_n) * s));
	} else {
		tmp = Math.exp(((0.5 * c_p) * s));
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if -s <= 1e-15:
		tmp = math.exp(((-0.5 * c_n) * s))
	else:
		tmp = math.exp(((0.5 * c_p) * s))
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (Float64(-s) <= 1e-15)
		tmp = exp(Float64(Float64(-0.5 * c_n) * s));
	else
		tmp = exp(Float64(Float64(0.5 * c_p) * s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (-s <= 1e-15)
		tmp = exp(((-0.5 * c_n) * s));
	else
		tmp = exp(((0.5 * c_p) * s));
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-s), 1e-15], N[Exp[N[(N[(-0.5 * c$95$n), $MachinePrecision] * s), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(0.5 * c$95$p), $MachinePrecision] * s), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 s) < 1.0000000000000001e-15

   1. Initial program 92.2%

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in s around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 99.6%

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

   11. Taylor expanded in c_p around 0

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

   13. Applied rewrites 99.6%

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

   if 1.0000000000000001e-15 < (neg.f64 s)

   1. Initial program 29.0%

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

   4. Applied rewrites 79.0%

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

   5. Taylor expanded in s around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 93.0%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 86.0%

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

   11. Taylor expanded in c_p around inf

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

   13. Applied rewrites 93.1%

       \[\leadsto e^{\left(0.5 \cdot c\_p\right) \cdot s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.7% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.7%

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

4. Applied rewrites 97.0%

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

5. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 98.8%

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

Alternative 5: 96.4% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.7%

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

4. Applied rewrites 97.0%

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

5. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 98.8%

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

11. Taylor expanded in c_p around 0

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

13. Applied rewrites 95.8%

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

Alternative 6: 93.9% accurate, 37.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.7%

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

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. lower-+.f64 N/A

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

   7. lower-exp.f64 N/A

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

   8. lower-neg.f64 N/A

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

   9. 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 90.4%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 90.0%

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

9. Taylor expanded in s around 0

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

11. Applied rewrites 94.7%

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

12. Taylor expanded in c_n around 0

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

14. Applied rewrites 94.7%

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

Alternative 7: 93.9% accurate, 49.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.7%

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

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. lower-+.f64 N/A

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

   7. lower-exp.f64 N/A

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

   8. lower-neg.f64 N/A

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

   9. 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 90.4%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 90.0%

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

9. Taylor expanded in s around 0

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

11. Applied rewrites 94.7%

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

12. Taylor expanded in c_n around 0

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

14. Applied rewrites 94.6%

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

Alternative 8: 93.9% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.7%

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

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. lower-+.f64 N/A

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

   7. lower-exp.f64 N/A

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

   8. lower-neg.f64 N/A

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

   9. 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 90.4%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 90.0%

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

9. Taylor expanded in s around 0

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

11. Applied rewrites 94.6%

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

Alternative 9: 93.9% 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 88.7%

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

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 93.0

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

5. Applied rewrites 93.0%

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

6. Taylor expanded in c_p around 0

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

8. Applied rewrites 94.2%

   \[\leadsto 1 \]
9. Add Preprocessing

Developer Target 1: 96.3% 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 2024319 
(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))))