NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.1% → 99.7%

Time: 10.4s

Alternatives: 2

Speedup: 1.9×

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: 78.1% 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: 99.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

   1. un-div-inv 75.8%

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

   2. difference-of-squares 89.9%

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

   3. associate-/r* 90.2%

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

   4. div-inv 90.2%

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

   5. metadata-eval 90.2%

      \[\leadsto \frac{\frac{\pi \cdot \color{blue}{0.5}}{b + a}}{b - a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   6. +-commutative 90.2%

      \[\leadsto \frac{\frac{\pi \cdot 0.5}{\color{blue}{a + b}}}{b - a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

4. Applied egg-rr 90.2%

   \[\leadsto \color{blue}{\frac{\frac{\pi \cdot 0.5}{a + b}}{b - a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
5. Step-by-step derivation

   1. associate-*l/ 99.7%

      \[\leadsto \color{blue}{\frac{\frac{\pi \cdot 0.5}{a + b} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b - a}} \]

6. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\frac{\pi \cdot 0.5}{a + b} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b - a}} \]
7. Step-by-step derivation

   1. clear-num 99.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{b - a}{\frac{\pi \cdot 0.5}{a + b} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}}} \]

   2. *-un-lft-identity 99.0%

      \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot \left(b - a\right)}}{\frac{\pi \cdot 0.5}{a + b} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}} \]

   3. times-frac 99.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\pi \cdot 0.5}{a + b}} \cdot \frac{b - a}{\frac{1}{a} - \frac{1}{b}}}} \]

   4. clear-num 99.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{a + b}{\pi \cdot 0.5}} \cdot \frac{b - a}{\frac{1}{a} - \frac{1}{b}}} \]

   5. *-un-lft-identity 99.0%

      \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot \left(a + b\right)}}{\pi \cdot 0.5} \cdot \frac{b - a}{\frac{1}{a} - \frac{1}{b}}} \]

   6. *-commutative 99.0%

      \[\leadsto \frac{1}{\frac{1 \cdot \left(a + b\right)}{\color{blue}{0.5 \cdot \pi}} \cdot \frac{b - a}{\frac{1}{a} - \frac{1}{b}}} \]

   7. times-frac 99.0%

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{0.5} \cdot \frac{a + b}{\pi}\right)} \cdot \frac{b - a}{\frac{1}{a} - \frac{1}{b}}} \]

   8. metadata-eval 99.0%

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

8. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{\frac{1}{\left(2 \cdot \frac{a + b}{\pi}\right) \cdot \frac{b - a}{\frac{1}{a} - \frac{1}{b}}}} \]
9. Step-by-step derivation

   1. associate-/r* 99.7%

      \[\leadsto \color{blue}{\frac{\frac{1}{2 \cdot \frac{a + b}{\pi}}}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}}} \]

   2. associate-/r* 99.7%

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{\frac{a + b}{\pi}}}}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}} \]

   3. metadata-eval 99.7%

      \[\leadsto \frac{\frac{\color{blue}{0.5}}{\frac{a + b}{\pi}}}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}} \]

   4. associate-/r/ 99.7%

      \[\leadsto \frac{\color{blue}{\frac{0.5}{a + b} \cdot \pi}}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}} \]

   5. associate-*l/ 99.7%

      \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot \pi}{a + b}}}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}} \]

   6. associate-*r/ 99.7%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{\pi}{a + b}}}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}} \]

10. Simplified 99.7%

    \[\leadsto \color{blue}{\frac{0.5 \cdot \frac{\pi}{a + b}}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}}} \]

11. Taylor expanded in b around 0 99.8%

    \[\leadsto \frac{0.5 \cdot \frac{\pi}{a + b}}{\color{blue}{a \cdot b}} \]
12. Step-by-step derivation

    1. *-commutative 99.8%

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

13. Simplified 99.8%

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

14. Final simplification 99.8%

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

Alternative 2: 65.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

   1. un-div-inv 75.8%

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

   2. difference-of-squares 89.9%

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

   3. associate-/r* 90.2%

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

   4. div-inv 90.2%

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

   5. metadata-eval 90.2%

      \[\leadsto \frac{\frac{\pi \cdot \color{blue}{0.5}}{b + a}}{b - a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   6. +-commutative 90.2%

      \[\leadsto \frac{\frac{\pi \cdot 0.5}{\color{blue}{a + b}}}{b - a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

4. Applied egg-rr 90.2%

   \[\leadsto \color{blue}{\frac{\frac{\pi \cdot 0.5}{a + b}}{b - a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
5. Step-by-step derivation

   1. associate-*l/ 99.7%

      \[\leadsto \color{blue}{\frac{\frac{\pi \cdot 0.5}{a + b} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b - a}} \]

6. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\frac{\pi \cdot 0.5}{a + b} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b - a}} \]

7. Taylor expanded in a around 0 69.6%

   \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{\pi}{a \cdot b}}}{b - a} \]
8. Step-by-step derivation

   1. associate-*r/ 69.6%

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

9. Simplified 69.6%

   \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot \pi}{a \cdot b}}}{b - a} \]
10. Step-by-step derivation

    1. associate-/l/ 69.5%

       \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{\left(b - a\right) \cdot \left(a \cdot b\right)}} \]

    2. times-frac 69.6%

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

11. Applied egg-rr 69.6%

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

Reproduce

herbie shell --seed 2024096 
(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))))