Harley's example

Percentage Accurate: 90.5% → 98.0%

Time: 1.4min

Alternatives: 5

Speedup: 835.0×

• could not determine a ground truth

  1. c_p = 2.00000000000000007e154
  2. c_n = 2
  3. t = 0.0
  4. s = 0.0

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: 90.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

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(((t * ((-0.5d0) * c_p)) + (s * (c_p * (0.5d0 + (s * (-0.125d0)))))))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return Math.exp(((t * (-0.5 * c_p)) + (s * (c_p * (0.5 + (s * -0.125))))));
}

Python

def code(c_p, c_n, t, s):
	return math.exp(((t * (-0.5 * c_p)) + (s * (c_p * (0.5 + (s * -0.125))))))

Julia

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

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = exp(((t * (-0.5 * c_p)) + (s * (c_p * (0.5 + (s * -0.125))))));
end

Wolfram

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

Derivation

1. Initial program 91.5%

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

   1. associate-/l/ 91.5%

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

3. Simplified 91.5%

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

5. Taylor expanded in c_n around 0 94.5%

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

   1. add-exp-log 94.5%

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

   2. log-div 94.5%

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

   3. log-pow 94.5%

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

   4. log-rec 94.5%

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

   5. log1p-define 94.5%

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

   6. log-pow 96.3%

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

   7. log-rec 96.3%

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

   8. log1p-define 96.3%

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

7. Applied egg-rr 96.3%

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

8. Taylor expanded in t around 0 96.4%

   \[\leadsto e^{\color{blue}{\left(-1 \cdot \left(c\_p \cdot \log \left(1 + e^{-s}\right)\right) + -0.5 \cdot \left(c\_p \cdot t\right)\right) - -1 \cdot \left(c\_p \cdot \log 2\right)}} \]
9. Step-by-step derivation

   1. +-commutative 96.4%

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

   2. associate--l+ 96.4%

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

   3. associate-*r* 96.4%

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

   4. *-commutative 96.4%

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

   5. associate-*r* 96.4%

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

   6. *-commutative 96.4%

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

   7. associate-*r* 96.4%

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

   8. *-commutative 96.4%

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

   9. distribute-lft-out-- 96.4%

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

   10. *-commutative 96.4%

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

   11. neg-mul-1 96.4%

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

   12. log1p-define 96.4%

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

10. Simplified 96.4%

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

11. Taylor expanded in s around 0 98.4%

    \[\leadsto e^{t \cdot \left(-0.5 \cdot c\_p\right) + \color{blue}{s \cdot \left(-0.125 \cdot \left(c\_p \cdot s\right) + 0.5 \cdot c\_p\right)}} \]
12. Step-by-step derivation

    1. associate-*r* 98.4%

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

    2. +-commutative 98.4%

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

    3. *-commutative 98.4%

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

    4. *-commutative 98.4%

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

    5. associate-*l* 98.4%

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

    6. distribute-lft-in 98.4%

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

    7. *-commutative 98.4%

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

13. Simplified 98.4%

    \[\leadsto e^{t \cdot \left(-0.5 \cdot c\_p\right) + \color{blue}{s \cdot \left(c\_p \cdot \left(0.5 + s \cdot -0.125\right)\right)}} \]
14. Add Preprocessing

Alternative 2: 96.5% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

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((c_p * ((t * (-0.5d0)) + (s * 0.5d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.5%

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

   1. associate-/l/ 91.5%

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

3. Simplified 91.5%

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

5. Taylor expanded in c_n around 0 94.5%

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

   1. add-exp-log 94.5%

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

   2. log-div 94.5%

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

   3. log-pow 94.5%

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

   4. log-rec 94.5%

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

   5. log1p-define 94.5%

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

   6. log-pow 96.3%

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

   7. log-rec 96.3%

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

   8. log1p-define 96.3%

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

7. Applied egg-rr 96.3%

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

8. Taylor expanded in t around 0 96.4%

   \[\leadsto e^{\color{blue}{\left(-1 \cdot \left(c\_p \cdot \log \left(1 + e^{-s}\right)\right) + -0.5 \cdot \left(c\_p \cdot t\right)\right) - -1 \cdot \left(c\_p \cdot \log 2\right)}} \]
9. Step-by-step derivation

   1. +-commutative 96.4%

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

   2. associate--l+ 96.4%

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

   3. associate-*r* 96.4%

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

   4. *-commutative 96.4%

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

   5. associate-*r* 96.4%

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

   6. *-commutative 96.4%

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

   7. associate-*r* 96.4%

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

   8. *-commutative 96.4%

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

   9. distribute-lft-out-- 96.4%

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

   10. *-commutative 96.4%

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

   11. neg-mul-1 96.4%

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

   12. log1p-define 96.4%

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

10. Simplified 96.4%

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

11. Taylor expanded in s around 0 97.2%

    \[\leadsto e^{\color{blue}{-0.5 \cdot \left(c\_p \cdot t\right) + 0.5 \cdot \left(c\_p \cdot s\right)}} \]
12. Step-by-step derivation

    1. *-commutative 97.2%

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

    2. associate-*r* 97.2%

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

    3. *-commutative 97.2%

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

    4. associate-*r* 97.2%

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

    5. distribute-rgt-out 97.2%

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

    6. *-commutative 97.2%

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

13. Simplified 97.2%

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

14. Final simplification 97.2%

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

Alternative 3: 94.0% accurate, 55.7× speedup

Math

\[\begin{array}{l} \\ 1 + s \cdot \left(c\_p \cdot \left(0.5 + s \cdot \left(-0.125 + c\_p \cdot 0.125\right)\right)\right) \end{array} \]

FPCore

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

C

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

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 + (s * (c_p * (0.5d0 + (s * ((-0.125d0) + (c_p * 0.125d0))))))
end function

Java

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

Python

def code(c_p, c_n, t, s):
	return 1.0 + (s * (c_p * (0.5 + (s * (-0.125 + (c_p * 0.125))))))

Julia

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

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = 1.0 + (s * (c_p * (0.5 + (s * (-0.125 + (c_p * 0.125))))));
end

Wolfram

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

Derivation

1. Initial program 91.5%

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

   1. associate-/l/ 91.5%

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

3. Simplified 91.5%

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

5. Taylor expanded in c_n around 0 94.5%

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

6. Taylor expanded in t around 0 94.5%

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

7. Taylor expanded in s around 0 95.3%

   \[\leadsto \color{blue}{1 + s \cdot \left(0.5 \cdot c\_p + s \cdot \left(-0.125 \cdot c\_p + 0.125 \cdot {c\_p}^{2}\right)\right)} \]

8. Taylor expanded in c_p around 0 95.3%

   \[\leadsto 1 + \color{blue}{c\_p \cdot \left(0.125 \cdot \left(c\_p \cdot {s}^{2}\right) + s \cdot \left(0.5 + -0.125 \cdot s\right)\right)} \]

10. Simplified 95.3%

    \[\leadsto 1 + \color{blue}{s \cdot \left(c\_p \cdot \left(0.5 + s \cdot \left(-0.125 + 0.125 \cdot c\_p\right)\right)\right)} \]

11. Final simplification 95.3%

    \[\leadsto 1 + s \cdot \left(c\_p \cdot \left(0.5 + s \cdot \left(-0.125 + c\_p \cdot 0.125\right)\right)\right) \]
12. Add Preprocessing

Alternative 4: 94.0% accurate, 119.3× speedup

Math

\[\begin{array}{l} \\ 1 + 0.5 \cdot \left(c\_p \cdot s\right) \end{array} \]

FPCore

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

C

double code(double c_p, double c_n, double t, double s) {
	return 1.0 + (0.5 * (c_p * 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 = 1.0d0 + (0.5d0 * (c_p * s))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return 1.0 + (0.5 * (c_p * s));
}

Python

def code(c_p, c_n, t, s):
	return 1.0 + (0.5 * (c_p * s))

Julia

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

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = 1.0 + (0.5 * (c_p * s));
end

Wolfram

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

Derivation

1. Initial program 91.5%

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

   1. associate-/l/ 91.5%

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

3. Simplified 91.5%

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

5. Taylor expanded in c_n around 0 94.5%

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

6. Taylor expanded in t around 0 94.5%

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

7. Taylor expanded in s around 0 95.3%

   \[\leadsto \color{blue}{1 + 0.5 \cdot \left(c\_p \cdot s\right)} \]
8. Step-by-step derivation

   1. *-commutative 95.3%

      \[\leadsto 1 + 0.5 \cdot \color{blue}{\left(s \cdot c\_p\right)} \]

9. Simplified 95.3%

   \[\leadsto \color{blue}{1 + 0.5 \cdot \left(s \cdot c\_p\right)} \]

10. Final simplification 95.3%

    \[\leadsto 1 + 0.5 \cdot \left(c\_p \cdot s\right) \]
11. Add Preprocessing

Alternative 5: 94.0% accurate, 835.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (c_p c_n t s) :precision binary64 1.0)

C

double code(double c_p, double c_n, double t, double s) {
	return 1.0;
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    code = 1.0d0
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return 1.0;
}

Python

def code(c_p, c_n, t, s):
	return 1.0

Julia

function code(c_p, c_n, t, s)
	return 1.0
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = 1.0;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := 1.0

Derivation

1. Initial program 91.5%

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

   1. associate-/l/ 91.5%

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

3. Simplified 91.5%

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

5. Taylor expanded in c_n around 0 94.5%

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

6. Taylor expanded in c_p around 0 95.2%

   \[\leadsto \color{blue}{1} \]
7. Add Preprocessing

Developer target: 96.3% accurate, 1.3× 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 2024090 
(FPCore (c_p c_n t s)
  :name "Harley's example"
  :precision binary64
  :pre (and (< 0.0 c_p) (< 0.0 c_n))

  :alt
  (* (pow (/ (+ 1.0 (exp (- t))) (+ 1.0 (exp (- s)))) c_p) (pow (/ (+ 1.0 (exp t)) (+ 1.0 (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))))