Harley's example

Percentage Accurate: 91.1% → 96.1%

Time: 2.3min

Alternatives: 5

Speedup: 835.0×

• could not determine a ground truth

  1. c_p = 1.9835975244251212e-27
  2. c_n = 2.6088635873265903e-30
  3. t = 8.647217716013217e+172
  4. s = -1.1836233073833514e-223

Specification

\[0 < c\_p \land 0 < c\_n\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq 10000000:\\ \;\;\;\;1 + c\_n \cdot \left(t \cdot \left(0.5 + t \cdot 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}}{0.5 \cdot \left(t \cdot c\_p\right) + {0.5}^{c\_p}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- s) 10000000.0)
   (+ 1.0 (* c_n (* t (+ 0.5 (* t 0.125)))))
   (/
    (pow (/ 1.0 (+ 1.0 (exp (- s)))) c_p)
    (+ (* 0.5 (* t c_p)) (pow 0.5 c_p)))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 10000000.0) {
		tmp = 1.0 + (c_n * (t * (0.5 + (t * 0.125))));
	} else {
		tmp = pow((1.0 / (1.0 + exp(-s))), c_p) / ((0.5 * (t * c_p)) + pow(0.5, c_p));
	}
	return tmp;
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    real(8) :: tmp
    if (-s <= 10000000.0d0) then
        tmp = 1.0d0 + (c_n * (t * (0.5d0 + (t * 0.125d0))))
    else
        tmp = ((1.0d0 / (1.0d0 + exp(-s))) ** c_p) / ((0.5d0 * (t * c_p)) + (0.5d0 ** c_p))
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 10000000.0) {
		tmp = 1.0 + (c_n * (t * (0.5 + (t * 0.125))));
	} else {
		tmp = Math.pow((1.0 / (1.0 + Math.exp(-s))), c_p) / ((0.5 * (t * c_p)) + Math.pow(0.5, c_p));
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if -s <= 10000000.0:
		tmp = 1.0 + (c_n * (t * (0.5 + (t * 0.125))))
	else:
		tmp = math.pow((1.0 / (1.0 + math.exp(-s))), c_p) / ((0.5 * (t * c_p)) + math.pow(0.5, c_p))
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (Float64(-s) <= 10000000.0)
		tmp = Float64(1.0 + Float64(c_n * Float64(t * Float64(0.5 + Float64(t * 0.125)))));
	else
		tmp = Float64((Float64(1.0 / Float64(1.0 + exp(Float64(-s)))) ^ c_p) / Float64(Float64(0.5 * Float64(t * c_p)) + (0.5 ^ c_p)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (-s <= 10000000.0)
		tmp = 1.0 + (c_n * (t * (0.5 + (t * 0.125))));
	else
		tmp = ((1.0 / (1.0 + exp(-s))) ^ c_p) / ((0.5 * (t * c_p)) + (0.5 ^ c_p));
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-s), 10000000.0], N[(1.0 + N[(c$95$n * N[(t * N[(0.5 + N[(t * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(1.0 / N[(1.0 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision] / N[(N[(0.5 * N[(t * c$95$p), $MachinePrecision]), $MachinePrecision] + N[Power[0.5, c$95$p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 s) < 1e7

   1. Initial program 94.0%

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

      1. associate-/l/ 94.0%

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

   3. Simplified 94.0%

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

   5. Taylor expanded in c_p around 0 96.8%

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

   6. Taylor expanded in s around 0 97.2%

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

   7. Taylor expanded in t around 0 97.9%

      \[\leadsto \color{blue}{1 + t \cdot \left(-1 \cdot \left(t \cdot \left(-0.25 \cdot {c\_n}^{2} + \left(-0.125 \cdot c\_n + 0.125 \cdot {c\_n}^{2}\right)\right)\right) - -0.5 \cdot c\_n\right)} \]

   8. Taylor expanded in c_n around 0 97.9%

      \[\leadsto 1 + \color{blue}{c\_n \cdot \left(t \cdot \left(0.5 + 0.125 \cdot t\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 97.9%

         \[\leadsto 1 + c\_n \cdot \left(t \cdot \left(0.5 + \color{blue}{t \cdot 0.125}\right)\right) \]

   10. Simplified 97.9%

       \[\leadsto 1 + \color{blue}{c\_n \cdot \left(t \cdot \left(0.5 + t \cdot 0.125\right)\right)} \]

   if 1e7 < (neg.f64 s)

   1. Initial program 50.0%

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

      1. associate-/l/ 50.0%

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

   3. Simplified 50.0%

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

   5. Taylor expanded in c_n around 0 50.0%

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

   6. Taylor expanded in t around 0 62.5%

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

   7. Taylor expanded in c_p around 0 100.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq 10000000:\\ \;\;\;\;1 + c\_n \cdot \left(t \cdot \left(0.5 + t \cdot 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}}{0.5 \cdot \left(t \cdot c\_p\right) + {0.5}^{c\_p}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(\frac{1}{2 + s \cdot \left(s \cdot 0.5 + -1\right)}\right)}^{c\_p}}{{0.5}^{c\_p}} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (/ (pow (/ 1.0 (+ 2.0 (* s (+ (* s 0.5) -1.0)))) c_p) (pow 0.5 c_p)))

C

double code(double c_p, double c_n, double t, double s) {
	return pow((1.0 / (2.0 + (s * ((s * 0.5) + -1.0)))), c_p) / pow(0.5, c_p);
}

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 / (2.0d0 + (s * ((s * 0.5d0) + (-1.0d0))))) ** c_p) / (0.5d0 ** c_p)
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return Math.pow((1.0 / (2.0 + (s * ((s * 0.5) + -1.0)))), c_p) / Math.pow(0.5, c_p);
}

Python

def code(c_p, c_n, t, s):
	return math.pow((1.0 / (2.0 + (s * ((s * 0.5) + -1.0)))), c_p) / math.pow(0.5, c_p)

Julia

function code(c_p, c_n, t, s)
	return Float64((Float64(1.0 / Float64(2.0 + Float64(s * Float64(Float64(s * 0.5) + -1.0)))) ^ c_p) / (0.5 ^ c_p))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = ((1.0 / (2.0 + (s * ((s * 0.5) + -1.0)))) ^ c_p) / (0.5 ^ c_p);
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[(N[Power[N[(1.0 / N[(2.0 + N[(s * N[(N[(s * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision] / N[Power[0.5, c$95$p], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.6%

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

   1. associate-/l/ 92.6%

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

3. Simplified 92.6%

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

5. Taylor expanded in c_n around 0 94.7%

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

6. Taylor expanded in s around 0 95.0%

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

7. Taylor expanded in t around 0 95.4%

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

8. Final simplification 95.4%

   \[\leadsto \frac{{\left(\frac{1}{2 + s \cdot \left(s \cdot 0.5 + -1\right)}\right)}^{c\_p}}{{0.5}^{c\_p}} \]
9. Add Preprocessing

Alternative 3: 94.0% accurate, 75.9× speedup

Math

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

FPCore

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

C

double code(double c_p, double c_n, double t, double s) {
	return 1.0 + (c_n * (t * (0.5 + (t * 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 + (c_n * (t * (0.5d0 + (t * 0.125d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.6%

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

   1. associate-/l/ 92.6%

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

3. Simplified 92.6%

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

5. Taylor expanded in c_p around 0 93.9%

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

6. Taylor expanded in s around 0 94.2%

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

7. Taylor expanded in t around 0 94.9%

   \[\leadsto \color{blue}{1 + t \cdot \left(-1 \cdot \left(t \cdot \left(-0.25 \cdot {c\_n}^{2} + \left(-0.125 \cdot c\_n + 0.125 \cdot {c\_n}^{2}\right)\right)\right) - -0.5 \cdot c\_n\right)} \]

8. Taylor expanded in c_n around 0 94.9%

   \[\leadsto 1 + \color{blue}{c\_n \cdot \left(t \cdot \left(0.5 + 0.125 \cdot t\right)\right)} \]
9. Step-by-step derivation

   1. *-commutative 94.9%

      \[\leadsto 1 + c\_n \cdot \left(t \cdot \left(0.5 + \color{blue}{t \cdot 0.125}\right)\right) \]

10. Simplified 94.9%

    \[\leadsto 1 + \color{blue}{c\_n \cdot \left(t \cdot \left(0.5 + t \cdot 0.125\right)\right)} \]
11. Add Preprocessing

Alternative 4: 94.0% accurate, 119.3× speedup

Math

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

FPCore

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

C

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

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_n * t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.6%

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

   1. associate-/l/ 92.6%

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

3. Simplified 92.6%

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

5. Taylor expanded in c_p around 0 93.9%

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

6. Taylor expanded in s around 0 94.2%

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

7. Taylor expanded in t around 0 94.9%

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

   1. *-commutative 94.9%

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

9. Simplified 94.9%

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

10. Final simplification 94.9%

    \[\leadsto 1 + 0.5 \cdot \left(c\_n \cdot t\right) \]
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 92.6%

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

   1. associate-/l/ 92.6%

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

3. Simplified 92.6%

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

5. Taylor expanded in c_n around 0 94.7%

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

6. Taylor expanded in c_p around 0 94.8%

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

Developer Target 1: 96.4% 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 2024170 
(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))))