Harley's example

Percentage Accurate: 90.7% → 99.5%

Time: 54.5s

Alternatives: 5

Speedup: 896.0×

• could not determine a ground truth

  1. c_p = 2.8915206688341904e-58
  2. c_n = 2780791456297161000.0
  3. t = -6.706043918917186e-134
  4. s = 7.896535456106975e+302

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: 90.7% 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, 6.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(c_p, c_n, t, s)
	return exp(fma(fma(-0.5, c_p, Float64(0.5 * c_n)), t, Float64(fma(-0.5, c_n, Float64(0.5 * c_p)) * 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] * t + N[(N[(-0.5 * c$95$n + N[(0.5 * c$95$p), $MachinePrecision]), $MachinePrecision] * s), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 90.2%

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

4. Applied rewrites 96.1%

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

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 97.2%

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

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 98.7%

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

11. Final simplification 98.7%

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

Alternative 2: 98.5% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 90.2%

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

4. Applied rewrites 96.1%

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

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 97.2%

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

8. Taylor expanded in s around inf

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

10. Applied rewrites 98.1%

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

11. Final simplification 98.1%

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

Alternative 3: 96.5% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.2%

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

4. Applied rewrites 96.1%

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

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 97.2%

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

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 98.7%

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

11. Taylor expanded in c_p around inf

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

13. Applied rewrites 94.1%

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

14. Final simplification 94.1%

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

Alternative 4: 93.7% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.2%

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

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

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

   8. lower-pow.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lower-exp.f64 N/A

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

   13. lower-neg.f64 91.4

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

5. Applied rewrites 91.4%

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

6. Taylor expanded in t around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-/.f64 N/A

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

8. Applied rewrites 90.7%

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

9. Taylor expanded in s around 0

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

11. Applied rewrites 91.4%

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

12. Taylor expanded in c_n around inf

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

14. Applied rewrites 91.5%

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

Alternative 5: 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 90.2%

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

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

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

   8. lower-pow.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lower-exp.f64 N/A

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

   13. lower-neg.f64 91.4

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

5. Applied rewrites 91.4%

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

6. Taylor expanded in c_p around 0

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

8. Applied rewrites 91.4%

   \[\leadsto 1 \]
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 2024235 
(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))))