Harley's example

Percentage Accurate: 90.9% → 99.5%

Time: 51.6s

Alternatives: 6

Speedup: 896.0×

• could not determine a ground truth

  1. c_p = 1.0005071132533099e-137
  2. c_n = 2.876152552104589e-147
  3. t = -4.399256437700166e+91
  4. s = 5.11231190070035e-20

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: 90.9% 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: 99.5% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.9%

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

4. Applied rewrites 96.9%

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

5. Taylor expanded in t around 0

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

   1. lower-fma.f64 N/A

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

   2. lower--.f64 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f64 N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-log.f64 N/A

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

7. Applied rewrites 99.1%

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

8. Taylor expanded in s around 0

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

10. Applied rewrites 99.8%

    \[\leadsto e^{\mathsf{fma}\left(s, \color{blue}{\mathsf{fma}\left(-0.5, c\_n, \mathsf{fma}\left(s, -0.125 \cdot \left(c\_n + c\_p\right), c\_p \cdot 0.5\right)\right)}, \mathsf{fma}\left(0.5, c\_n, -0.5 \cdot c\_p\right) \cdot t\right)} \]

11. Final simplification 99.8%

    \[\leadsto e^{\mathsf{fma}\left(s, \mathsf{fma}\left(-0.5, c\_n, \mathsf{fma}\left(s, -0.125 \cdot \left(c\_p + c\_n\right), c\_p \cdot 0.5\right)\right), t \cdot \mathsf{fma}\left(0.5, c\_n, c\_p \cdot -0.5\right)\right)} \]
12. Add Preprocessing

Alternative 2: 99.4% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.9%

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

4. Applied rewrites 96.9%

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

5. Taylor expanded in t around 0

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

   1. lower-fma.f64 N/A

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

   2. lower--.f64 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f64 N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-log.f64 N/A

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

7. Applied rewrites 99.1%

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

8. Taylor expanded in s 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 99.7%

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

11. Final simplification 99.7%

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

Alternative 3: 98.7% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.9%

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

4. Applied rewrites 96.9%

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

5. Taylor expanded in t around 0

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

   1. lower-fma.f64 N/A

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

   2. lower--.f64 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f64 N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-log.f64 N/A

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

7. Applied rewrites 99.1%

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

8. Taylor expanded in s around 0

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

10. Applied rewrites 99.8%

    \[\leadsto e^{\mathsf{fma}\left(s, \color{blue}{\mathsf{fma}\left(-0.5, c\_n, \mathsf{fma}\left(s, -0.125 \cdot \left(c\_n + c\_p\right), c\_p \cdot 0.5\right)\right)}, \mathsf{fma}\left(0.5, c\_n, -0.5 \cdot c\_p\right) \cdot t\right)} \]

11. Taylor expanded in c_n around inf

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

13. Applied rewrites 99.1%

    \[\leadsto e^{c\_n \cdot \mathsf{fma}\left(s, \color{blue}{\mathsf{fma}\left(-0.125, s, -0.5\right)}, 0.5 \cdot t\right)} \]

14. Final simplification 99.1%

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

Alternative 4: 98.6% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.9%

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

4. Applied rewrites 96.9%

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

5. Taylor expanded in t around 0

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

   1. lower-fma.f64 N/A

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

   2. lower--.f64 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f64 N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-log.f64 N/A

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

7. Applied rewrites 99.1%

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

8. Taylor expanded in s 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 99.7%

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

11. Taylor expanded in t around 0

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

13. Applied rewrites 99.0%

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

Alternative 5: 95.1% accurate, 8.1× speedup

Math

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

FPCore

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

C

double code(double c_p, double c_n, double t, double s) {
	return exp((0.5 * (t * 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 = exp((0.5d0 * (t * c_n)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.9%

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

4. Applied rewrites 96.9%

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

5. Taylor expanded in t around 0

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

   1. lower-fma.f64 N/A

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

   2. lower--.f64 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f64 N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-log.f64 N/A

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

7. Applied rewrites 99.1%

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

8. Taylor expanded in s around 0

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

10. Applied rewrites 96.5%

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

11. Taylor expanded in c_n around inf

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

13. Applied rewrites 96.5%

    \[\leadsto e^{0.5 \cdot \left(c\_n \cdot \color{blue}{t}\right)} \]

14. Final simplification 96.5%

    \[\leadsto e^{0.5 \cdot \left(t \cdot c\_n\right)} \]
15. Add Preprocessing

Alternative 6: 93.8% 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 91.9%

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

3. Taylor expanded in c_n around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-neg.f64 93.4

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

5. Applied rewrites 93.4%

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

6. Taylor expanded in c_p around 0

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

8. Applied rewrites 95.4%

   \[\leadsto 1 \]
9. Add Preprocessing

Developer Target 1: 96.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024232 
(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))))