Harley's example

Percentage Accurate: 91.3% → 99.5%

Time: 2.5min

Alternatives: 3

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: 91.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

4. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{e^{\left(c\_n \cdot \mathsf{log1p}\left(\frac{1}{-1 + \frac{-1}{e^{s}}}\right) - c\_n \cdot \mathsf{log1p}\left(\frac{1}{-1 + \frac{-1}{e^{t}}}\right)\right) + \left(\left(-\mathsf{log1p}\left(e^{0 - s}\right) \cdot c\_p\right) + \mathsf{log1p}\left(e^{0 - t}\right) \cdot c\_p\right)}} \]

5. Taylor expanded in t around 0

   \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(\left(c\_n \cdot \log \left(1 - \frac{1}{1 + \frac{1}{e^{s}}}\right) + \left(c\_p \cdot \log 2 + t \cdot \left(\left(\frac{-1}{2} \cdot c\_p + t \cdot \left(\frac{1}{8} \cdot c\_p - \frac{-1}{8} \cdot c\_n\right)\right) - \frac{-1}{2} \cdot c\_n\right)\right)\right) - \left(c\_n \cdot \log \frac{1}{2} + c\_p \cdot \log \left(1 + e^{\mathsf{neg}\left(s\right)}\right)\right)\right)}\right) \]

7. Simplified 99.2%

   \[\leadsto e^{\color{blue}{\left(\left(c\_p \cdot \log 2 + t \cdot \left(t \cdot \left(0.125 \cdot \left(c\_p + c\_n\right)\right) + -0.5 \cdot \left(c\_p - c\_n\right)\right)\right) + c\_n \cdot \left(\mathsf{log1p}\left(\frac{-1}{1 + e^{0 - s}}\right) - \log 0.5\right)\right) - c\_p \cdot \mathsf{log1p}\left(e^{0 - s}\right)}} \]

8. Taylor expanded in s around 0

   \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(s \cdot \left(\frac{-1}{2} \cdot c\_n - \frac{-1}{2} \cdot c\_p\right) + t \cdot \left(\frac{-1}{2} \cdot \left(c\_p - c\_n\right) + \frac{1}{8} \cdot \left(t \cdot \left(c\_n + c\_p\right)\right)\right)\right)}\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(t \cdot \left(\frac{-1}{2} \cdot \left(c\_p - c\_n\right) + \frac{1}{8} \cdot \left(t \cdot \left(c\_n + c\_p\right)\right)\right) + s \cdot \left(\frac{-1}{2} \cdot c\_n - \frac{-1}{2} \cdot c\_p\right)\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(t \cdot \left(\frac{-1}{2} \cdot \left(c\_p - c\_n\right) + \frac{1}{8} \cdot \left(t \cdot \left(c\_n + c\_p\right)\right)\right)\right), \left(s \cdot \left(\frac{-1}{2} \cdot c\_n - \frac{-1}{2} \cdot c\_p\right)\right)\right)\right) \]

10. Simplified 99.2%

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

11. Taylor expanded in t around 0

    \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \left(s \cdot \left(c\_n - c\_p\right)\right) + \frac{1}{2} \cdot \left(t \cdot \left(c\_n - c\_p\right)\right)\right)}\right) \]
12. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \mathsf{exp.f64}\left(\left(\left(\frac{-1}{2} \cdot s\right) \cdot \left(c\_n - c\_p\right) + \frac{1}{2} \cdot \left(t \cdot \left(c\_n - c\_p\right)\right)\right)\right) \]

    2. associate-*r* N/A

       \[\leadsto \mathsf{exp.f64}\left(\left(\left(\frac{-1}{2} \cdot s\right) \cdot \left(c\_n - c\_p\right) + \left(\frac{1}{2} \cdot t\right) \cdot \left(c\_n - c\_p\right)\right)\right) \]

    3. distribute-rgt-out N/A

       \[\leadsto \mathsf{exp.f64}\left(\left(\left(c\_n - c\_p\right) \cdot \left(\frac{-1}{2} \cdot s + \frac{1}{2} \cdot t\right)\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left(c\_n - c\_p\right), \left(\frac{-1}{2} \cdot s + \frac{1}{2} \cdot t\right)\right)\right) \]

    5. --lowering--.f64 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(c\_n, c\_p\right), \left(\frac{-1}{2} \cdot s + \frac{1}{2} \cdot t\right)\right)\right) \]

    6. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(c\_n, c\_p\right), \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot s\right), \left(\frac{1}{2} \cdot t\right)\right)\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(c\_n, c\_p\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, s\right), \left(\frac{1}{2} \cdot t\right)\right)\right)\right) \]

    8. *-lowering-*.f64 99.9%

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(c\_n, c\_p\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, s\right), \mathsf{*.f64}\left(\frac{1}{2}, t\right)\right)\right)\right) \]

13. Simplified 99.9%

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

Alternative 2: 98.5% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

4. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{e^{\left(c\_n \cdot \mathsf{log1p}\left(\frac{1}{-1 + \frac{-1}{e^{s}}}\right) - c\_n \cdot \mathsf{log1p}\left(\frac{1}{-1 + \frac{-1}{e^{t}}}\right)\right) + \left(\left(-\mathsf{log1p}\left(e^{0 - s}\right) \cdot c\_p\right) + \mathsf{log1p}\left(e^{0 - t}\right) \cdot c\_p\right)}} \]

5. Taylor expanded in t around 0

   \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(\left(c\_n \cdot \log \left(1 - \frac{1}{1 + \frac{1}{e^{s}}}\right) + \left(c\_p \cdot \log 2 + t \cdot \left(\left(\frac{-1}{2} \cdot c\_p + t \cdot \left(\frac{1}{8} \cdot c\_p - \frac{-1}{8} \cdot c\_n\right)\right) - \frac{-1}{2} \cdot c\_n\right)\right)\right) - \left(c\_n \cdot \log \frac{1}{2} + c\_p \cdot \log \left(1 + e^{\mathsf{neg}\left(s\right)}\right)\right)\right)}\right) \]

7. Simplified 99.2%

   \[\leadsto e^{\color{blue}{\left(\left(c\_p \cdot \log 2 + t \cdot \left(t \cdot \left(0.125 \cdot \left(c\_p + c\_n\right)\right) + -0.5 \cdot \left(c\_p - c\_n\right)\right)\right) + c\_n \cdot \left(\mathsf{log1p}\left(\frac{-1}{1 + e^{0 - s}}\right) - \log 0.5\right)\right) - c\_p \cdot \mathsf{log1p}\left(e^{0 - s}\right)}} \]

8. Taylor expanded in s around 0

   \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(s \cdot \left(\frac{-1}{2} \cdot c\_n - \frac{-1}{2} \cdot c\_p\right) + t \cdot \left(\frac{-1}{2} \cdot \left(c\_p - c\_n\right) + \frac{1}{8} \cdot \left(t \cdot \left(c\_n + c\_p\right)\right)\right)\right)}\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(t \cdot \left(\frac{-1}{2} \cdot \left(c\_p - c\_n\right) + \frac{1}{8} \cdot \left(t \cdot \left(c\_n + c\_p\right)\right)\right) + s \cdot \left(\frac{-1}{2} \cdot c\_n - \frac{-1}{2} \cdot c\_p\right)\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(t \cdot \left(\frac{-1}{2} \cdot \left(c\_p - c\_n\right) + \frac{1}{8} \cdot \left(t \cdot \left(c\_n + c\_p\right)\right)\right)\right), \left(s \cdot \left(\frac{-1}{2} \cdot c\_n - \frac{-1}{2} \cdot c\_p\right)\right)\right)\right) \]

10. Simplified 99.2%

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

11. Taylor expanded in t around 0

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

    1. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{exp.f64}\left(\left(\frac{-1}{2} \cdot \left(s \cdot \left(c\_n - c\_p\right)\right)\right)\right) \]

    2. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(s \cdot \left(c\_n - c\_p\right)\right)\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(s, \left(c\_n - c\_p\right)\right)\right)\right) \]

    4. --lowering--.f64 99.4%

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(s, \mathsf{\_.f64}\left(c\_n, c\_p\right)\right)\right)\right) \]

13. Simplified 99.4%

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

14. Final simplification 99.4%

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

Alternative 3: 94.5% 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 94.5%

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

3. Taylor expanded in c_p around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}\right), \color{blue}{\left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)}\right) \]

   2. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right), c\_n\right), \left({\color{blue}{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}}^{c\_n}\right)\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)\right), c\_n\right), \left({\left(\color{blue}{1} - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)\right), c\_n\right), \left({\left(\color{blue}{1} - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

   5. distribute-neg-frac N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \left(1 + e^{\mathsf{neg}\left(s\right)}\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

   8. neg-mul-1 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \left(1 + e^{-1 \cdot s}\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \left(e^{-1 \cdot s}\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

   10. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(-1 \cdot s\right)\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

   11. neg-mul-1 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(s\right)\right)\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

   12. neg-sub0 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(0 - s\right)\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

   13. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, s\right)\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

   14. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, s\right)\right)\right)\right)\right), c\_n\right), \mathsf{pow.f64}\left(\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right), \color{blue}{c\_n}\right)\right) \]

5. Simplified 94.6%

   \[\leadsto \color{blue}{\frac{{\left(1 + \frac{-1}{1 + e^{0 - s}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{0 - t}}\right)}^{c\_n}}} \]

6. Taylor expanded in c_n around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 94.8%

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

Developer Target 1: 96.2% 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 2024138 
(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))))