NMSE Section 6.1 mentioned, B

Percentage Accurate: 77.6% → 99.6%

Time: 11.1s

Alternatives: 10

Speedup: 2.4×

Specification

Math

\[\begin{array}{l} \\ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))

C

double code(double a, double b) {
	return ((((double) M_PI) / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}

Java

public static double code(double a, double b) {
	return ((Math.PI / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}

Python

def code(a, b):
	return ((math.pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b))

Julia

function code(a, b)
	return Float64(Float64(Float64(pi / 2.0) * Float64(1.0 / Float64(Float64(b * b) - Float64(a * a)))) * Float64(Float64(1.0 / a) - Float64(1.0 / b)))
end

MATLAB

function tmp = code(a, b)
	tmp = ((pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
end

Wolfram

code[a_, b_] := N[(N[(N[(Pi / 2.0), $MachinePrecision] * N[(1.0 / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))

C

double code(double a, double b) {
	return ((((double) M_PI) / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}

Java

public static double code(double a, double b) {
	return ((Math.PI / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}

Python

def code(a, b):
	return ((math.pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b))

Julia

function code(a, b)
	return Float64(Float64(Float64(pi / 2.0) * Float64(1.0 / Float64(Float64(b * b) - Float64(a * a)))) * Float64(Float64(1.0 / a) - Float64(1.0 / b)))
end

MATLAB

function tmp = code(a, b)
	tmp = ((pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
end

Wolfram

code[a_, b_] := N[(N[(N[(Pi / 2.0), $MachinePrecision] * N[(1.0 / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{b + a} \cdot \frac{\frac{0.5}{b}}{a} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (* (/ PI (+ b a)) (/ (/ 0.5 b) a)))

C

double code(double a, double b) {
	return (((double) M_PI) / (b + a)) * ((0.5 / b) / a);
}

Java

public static double code(double a, double b) {
	return (Math.PI / (b + a)) * ((0.5 / b) / a);
}

Python

def code(a, b):
	return (math.pi / (b + a)) * ((0.5 / b) / a)

Julia

function code(a, b)
	return Float64(Float64(pi / Float64(b + a)) * Float64(Float64(0.5 / b) / a))
end

MATLAB

function tmp = code(a, b)
	tmp = (pi / (b + a)) * ((0.5 / b) / a);
end

Wolfram

code[a_, b_] := N[(N[(Pi / N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.7%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   3. lift-/.f64 N/A

      \[\leadsto \left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \color{blue}{\frac{1}{b \cdot b - a \cdot a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   4. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b \cdot b - a \cdot a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   6. div-inv N/A

      \[\leadsto \frac{\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   7. lift--.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}{\color{blue}{b \cdot b - a \cdot a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}{\color{blue}{b \cdot b} - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}{b \cdot b - \color{blue}{a \cdot a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   10. difference-of-squares N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   11. times-frac N/A

       \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{b + a} \cdot \frac{\frac{1}{2}}{b - a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   12. associate-*l* N/A

       \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{b + a} \cdot \left(\frac{\frac{1}{2}}{b - a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right)} \]

   13. lower-*.f64 N/A

       \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{b + a} \cdot \left(\frac{\frac{1}{2}}{b - a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right)} \]

   14. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{b + a}} \cdot \left(\frac{\frac{1}{2}}{b - a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right) \]

   15. lower-+.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{b + a}} \cdot \left(\frac{\frac{1}{2}}{b - a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{b + a} \cdot \color{blue}{\left(\frac{\frac{1}{2}}{b - a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right)} \]

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{\pi}{b + a} \cdot \left(\frac{0.5}{b - a} \cdot \frac{b - a}{b \cdot a}\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{b + a} \cdot \color{blue}{\left(\frac{\frac{1}{2}}{b - a} \cdot \frac{b - a}{b \cdot a}\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{b + a} \cdot \left(\frac{\frac{1}{2}}{b - a} \cdot \color{blue}{\frac{b - a}{b \cdot a}}\right) \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{b + a} \cdot \color{blue}{\frac{\frac{\frac{1}{2}}{b - a} \cdot \left(b - a\right)}{b \cdot a}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{b + a} \cdot \frac{\frac{\frac{1}{2}}{b - a} \cdot \left(b - a\right)}{\color{blue}{b \cdot a}} \]

   5. associate-/r* N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{b + a} \cdot \color{blue}{\frac{\frac{\frac{\frac{1}{2}}{b - a} \cdot \left(b - a\right)}{b}}{a}} \]

   6. lift-/.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{b + a} \cdot \frac{\frac{\color{blue}{\frac{\frac{1}{2}}{b - a}} \cdot \left(b - a\right)}{b}}{a} \]

   7. div-inv N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{b + a} \cdot \frac{\frac{\color{blue}{\left(\frac{1}{2} \cdot \frac{1}{b - a}\right)} \cdot \left(b - a\right)}{b}}{a} \]

   8. associate-*l* N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{b + a} \cdot \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(\frac{1}{b - a} \cdot \left(b - a\right)\right)}}{b}}{a} \]

   9. inv-pow N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{b + a} \cdot \frac{\frac{\frac{1}{2} \cdot \left(\color{blue}{{\left(b - a\right)}^{-1}} \cdot \left(b - a\right)\right)}{b}}{a} \]

   10. pow-plus N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{b + a} \cdot \frac{\frac{\frac{1}{2} \cdot \color{blue}{{\left(b - a\right)}^{\left(-1 + 1\right)}}}{b}}{a} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{b + a} \cdot \frac{\frac{\frac{1}{2} \cdot {\left(b - a\right)}^{\color{blue}{0}}}{b}}{a} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{b + a} \cdot \frac{\frac{\frac{1}{2} \cdot \color{blue}{1}}{b}}{a} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{b + a} \cdot \frac{\frac{\color{blue}{\frac{1}{2}}}{b}}{a} \]

   14. lift-/.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{b + a} \cdot \frac{\color{blue}{\frac{\frac{1}{2}}{b}}}{a} \]

   15. lower-/.f64 99.7

       \[\leadsto \frac{\pi}{b + a} \cdot \color{blue}{\frac{\frac{0.5}{b}}{a}} \]

6. Applied rewrites 99.7%

   \[\leadsto \frac{\pi}{b + a} \cdot \color{blue}{\frac{\frac{0.5}{b}}{a}} \]
7. Add Preprocessing

Alternative 2: 99.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\pi}{b + a}}{2 \cdot \left(b \cdot a\right)} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ (/ PI (+ b a)) (* 2.0 (* b a))))

C

double code(double a, double b) {
	return (((double) M_PI) / (b + a)) / (2.0 * (b * a));
}

Java

public static double code(double a, double b) {
	return (Math.PI / (b + a)) / (2.0 * (b * a));
}

Python

def code(a, b):
	return (math.pi / (b + a)) / (2.0 * (b * a))

Julia

function code(a, b)
	return Float64(Float64(pi / Float64(b + a)) / Float64(2.0 * Float64(b * a)))
end

MATLAB

function tmp = code(a, b)
	tmp = (pi / (b + a)) / (2.0 * (b * a));
end

Wolfram

code[a_, b_] := N[(N[(Pi / N[(b + a), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.6%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   3. lift-/.f64 N/A

      \[\leadsto \left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \color{blue}{\frac{1}{b \cdot b - a \cdot a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   4. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b \cdot b - a \cdot a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   6. div-inv N/A

      \[\leadsto \frac{\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   7. lift--.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}{\color{blue}{b \cdot b - a \cdot a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}{\color{blue}{b \cdot b} - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}{b \cdot b - \color{blue}{a \cdot a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   10. difference-of-squares N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   11. times-frac N/A

       \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{b + a} \cdot \frac{\frac{1}{2}}{b - a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   12. associate-*l* N/A

       \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{b + a} \cdot \left(\frac{\frac{1}{2}}{b - a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right)} \]

   13. lower-*.f64 N/A

       \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{b + a} \cdot \left(\frac{\frac{1}{2}}{b - a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right)} \]

   14. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{b + a}} \cdot \left(\frac{\frac{1}{2}}{b - a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right) \]

   15. lower-+.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{b + a}} \cdot \left(\frac{\frac{1}{2}}{b - a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{b + a} \cdot \color{blue}{\left(\frac{\frac{1}{2}}{b - a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right)} \]

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\pi}{b + a} \cdot \left(\frac{0.5}{b - a} \cdot \frac{b - a}{b \cdot a}\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{b + a} \cdot \left(\frac{\frac{1}{2}}{b - a} \cdot \frac{b - a}{b \cdot a}\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{b + a}} \cdot \left(\frac{\frac{1}{2}}{b - a} \cdot \frac{b - a}{b \cdot a}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{b + a} \cdot \color{blue}{\left(\frac{\frac{1}{2}}{b - a} \cdot \frac{b - a}{b \cdot a}\right)} \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{b + a} \cdot \left(\color{blue}{\frac{\frac{1}{2}}{b - a}} \cdot \frac{b - a}{b \cdot a}\right) \]

   5. associate-*l/ N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{b + a} \cdot \color{blue}{\frac{\frac{1}{2} \cdot \frac{b - a}{b \cdot a}}{b - a}} \]

   6. frac-times N/A

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} \cdot \frac{b - a}{b \cdot a}\right)}{\left(b + a\right) \cdot \left(b - a\right)}} \]

   7. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \frac{b - a}{b \cdot a}}}{\left(b + a\right) \cdot \left(b - a\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)} \cdot \frac{b - a}{b \cdot a}}{\left(b + a\right) \cdot \left(b - a\right)} \]

   9. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{b - a}{b \cdot a} \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)}}{\left(b + a\right) \cdot \left(b - a\right)} \]

   10. lift-/.f64 N/A

       \[\leadsto \frac{\color{blue}{\frac{b - a}{b \cdot a}} \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)}{\left(b + a\right) \cdot \left(b - a\right)} \]

   11. associate-*l/ N/A

       \[\leadsto \frac{\color{blue}{\frac{\left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)}{b \cdot a}}}{\left(b + a\right) \cdot \left(b - a\right)} \]

   12. lift-+.f64 N/A

       \[\leadsto \frac{\frac{\left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)}{b \cdot a}}{\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)} \]

   13. lift--.f64 N/A

       \[\leadsto \frac{\frac{\left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)}{b \cdot a}}{\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}} \]

   14. difference-of-squares N/A

       \[\leadsto \frac{\frac{\left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)}{b \cdot a}}{\color{blue}{b \cdot b - a \cdot a}} \]

   15. associate-/l/ N/A

       \[\leadsto \color{blue}{\frac{\left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)}{\left(b \cdot b - a \cdot a\right) \cdot \left(b \cdot a\right)}} \]

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{\frac{\pi}{b + a}}{2 \cdot \left(b \cdot a\right)}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024223 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))