bug323 (missed optimization)

Percentage Accurate: 6.9% → 10.4%

Time: 18.5s

Alternatives: 6

Speedup: 1.0×

Specification

\[0 \leq x \land x \leq 0.5\]

Math

\[\begin{array}{l} \\ \cos^{-1} \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (acos (- 1.0 x)))

C

double code(double x) {
	return acos((1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = acos((1.0d0 - x))
end function

Java

public static double code(double x) {
	return Math.acos((1.0 - x));
}

Python

def code(x):
	return math.acos((1.0 - x))

Julia

function code(x)
	return acos(Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = acos((1.0 - x));
end

Wolfram

code[x_] := N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]

Initial Program: 6.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (acos (- 1.0 x)))

C

double code(double x) {
	return acos((1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = acos((1.0d0 - x))
end function

Java

public static double code(double x) {
	return Math.acos((1.0 - x));
}

Python

def code(x):
	return math.acos((1.0 - x))

Julia

function code(x)
	return acos(Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = acos((1.0 - x));
end

Wolfram

code[x_] := N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]

Alternative 1: 10.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \pi \cdot 0.5 - {\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{1.5}}\right)}^{2} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (* PI 0.5) (pow (cbrt (pow (asin (- 1.0 x)) 1.5)) 2.0)))

C

double code(double x) {
	return (((double) M_PI) * 0.5) - pow(cbrt(pow(asin((1.0 - x)), 1.5)), 2.0);
}

Java

public static double code(double x) {
	return (Math.PI * 0.5) - Math.pow(Math.cbrt(Math.pow(Math.asin((1.0 - x)), 1.5)), 2.0);
}

Julia

function code(x)
	return Float64(Float64(pi * 0.5) - (cbrt((asin(Float64(1.0 - x)) ^ 1.5)) ^ 2.0))
end

Wolfram

code[x_] := N[(N[(Pi * 0.5), $MachinePrecision] - N[Power[N[Power[N[Power[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.3%

   \[\cos^{-1} \left(1 - x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. acos-asin 7.3%

      \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)} \]

   2. sub-neg 7.3%

      \[\leadsto \color{blue}{\frac{\pi}{2} + \left(-\sin^{-1} \left(1 - x\right)\right)} \]

   3. div-inv 7.3%

      \[\leadsto \color{blue}{\pi \cdot \frac{1}{2}} + \left(-\sin^{-1} \left(1 - x\right)\right) \]

   4. metadata-eval 7.3%

      \[\leadsto \pi \cdot \color{blue}{0.5} + \left(-\sin^{-1} \left(1 - x\right)\right) \]

4. Applied egg-rr 7.3%

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

   1. sub-neg 7.3%

      \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)} \]

6. Simplified 7.3%

   \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)} \]
7. Step-by-step derivation

   1. add-sqr-sqrt 10.5%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{\sqrt{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}} \]

   2. pow2 10.5%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2}} \]

8. Applied egg-rr 10.5%

   \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2}} \]
9. Step-by-step derivation

   1. add-cbrt-cube 10.5%

      \[\leadsto \pi \cdot 0.5 - {\color{blue}{\left(\sqrt[3]{\left(\sqrt{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt{\sin^{-1} \left(1 - x\right)}}\right)}}^{2} \]

   2. add-sqr-sqrt 5.5%

      \[\leadsto \pi \cdot 0.5 - {\left(\sqrt[3]{\color{blue}{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}}\right)}^{2} \]

   3. pow1 5.5%

      \[\leadsto \pi \cdot 0.5 - {\left(\sqrt[3]{\color{blue}{{\sin^{-1} \left(1 - x\right)}^{1}} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}}\right)}^{2} \]

   4. pow1/2 5.5%

      \[\leadsto \pi \cdot 0.5 - {\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{1} \cdot \color{blue}{{\sin^{-1} \left(1 - x\right)}^{0.5}}}\right)}^{2} \]

   5. pow-prod-up 10.5%

      \[\leadsto \pi \cdot 0.5 - {\left(\sqrt[3]{\color{blue}{{\sin^{-1} \left(1 - x\right)}^{\left(1 + 0.5\right)}}}\right)}^{2} \]

   6. metadata-eval 10.5%

      \[\leadsto \pi \cdot 0.5 - {\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{\color{blue}{1.5}}}\right)}^{2} \]

10. Applied egg-rr 10.5%

    \[\leadsto \pi \cdot 0.5 - {\color{blue}{\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{1.5}}\right)}}^{2} \]

11. Final simplification 10.5%

    \[\leadsto \pi \cdot 0.5 - {\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{1.5}}\right)}^{2} \]
12. Add Preprocessing

Alternative 2: 10.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \pi \cdot 0.5 - {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (* PI 0.5) (pow (cbrt (asin (- 1.0 x))) 3.0)))

C

double code(double x) {
	return (((double) M_PI) * 0.5) - pow(cbrt(asin((1.0 - x))), 3.0);
}

Java

public static double code(double x) {
	return (Math.PI * 0.5) - Math.pow(Math.cbrt(Math.asin((1.0 - x))), 3.0);
}

Julia

function code(x)
	return Float64(Float64(pi * 0.5) - (cbrt(asin(Float64(1.0 - x))) ^ 3.0))
end

Wolfram

code[x_] := N[(N[(Pi * 0.5), $MachinePrecision] - N[Power[N[Power[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.3%

   \[\cos^{-1} \left(1 - x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. acos-asin 7.3%

      \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)} \]

   2. sub-neg 7.3%

      \[\leadsto \color{blue}{\frac{\pi}{2} + \left(-\sin^{-1} \left(1 - x\right)\right)} \]

   3. div-inv 7.3%

      \[\leadsto \color{blue}{\pi \cdot \frac{1}{2}} + \left(-\sin^{-1} \left(1 - x\right)\right) \]

   4. metadata-eval 7.3%

      \[\leadsto \pi \cdot \color{blue}{0.5} + \left(-\sin^{-1} \left(1 - x\right)\right) \]

4. Applied egg-rr 7.3%

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

   1. sub-neg 7.3%

      \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)} \]

6. Simplified 7.3%

   \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)} \]
7. Step-by-step derivation

   1. add-cube-cbrt 10.5%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}} \]

   2. pow3 10.5%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \]

8. Applied egg-rr 10.5%

   \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \]

9. Final simplification 10.5%

   \[\leadsto \pi \cdot 0.5 - {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3} \]
10. Add Preprocessing

Alternative 3: 10.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \pi \cdot {\left(\sqrt{0.5}\right)}^{2} - \sin^{-1} \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (* PI (pow (sqrt 0.5) 2.0)) (asin (- 1.0 x))))

C

double code(double x) {
	return (((double) M_PI) * pow(sqrt(0.5), 2.0)) - asin((1.0 - x));
}

Java

public static double code(double x) {
	return (Math.PI * Math.pow(Math.sqrt(0.5), 2.0)) - Math.asin((1.0 - x));
}

Python

def code(x):
	return (math.pi * math.pow(math.sqrt(0.5), 2.0)) - math.asin((1.0 - x))

Julia

function code(x)
	return Float64(Float64(pi * (sqrt(0.5) ^ 2.0)) - asin(Float64(1.0 - x)))
end

MATLAB

function tmp = code(x)
	tmp = (pi * (sqrt(0.5) ^ 2.0)) - asin((1.0 - x));
end

Wolfram

code[x_] := N[(N[(Pi * N[Power[N[Sqrt[0.5], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.3%

   \[\cos^{-1} \left(1 - x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. acos-asin 7.3%

      \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)} \]

   2. add-sqr-sqrt 5.5%

      \[\leadsto \color{blue}{\sqrt{\frac{\pi}{2}} \cdot \sqrt{\frac{\pi}{2}}} - \sin^{-1} \left(1 - x\right) \]

   3. fma-neg 5.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{\pi}{2}}, \sqrt{\frac{\pi}{2}}, -\sin^{-1} \left(1 - x\right)\right)} \]

   4. div-inv 5.5%

      \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\pi \cdot \frac{1}{2}}}, \sqrt{\frac{\pi}{2}}, -\sin^{-1} \left(1 - x\right)\right) \]

   5. metadata-eval 5.5%

      \[\leadsto \mathsf{fma}\left(\sqrt{\pi \cdot \color{blue}{0.5}}, \sqrt{\frac{\pi}{2}}, -\sin^{-1} \left(1 - x\right)\right) \]

   6. div-inv 5.5%

      \[\leadsto \mathsf{fma}\left(\sqrt{\pi \cdot 0.5}, \sqrt{\color{blue}{\pi \cdot \frac{1}{2}}}, -\sin^{-1} \left(1 - x\right)\right) \]

   7. metadata-eval 5.5%

      \[\leadsto \mathsf{fma}\left(\sqrt{\pi \cdot 0.5}, \sqrt{\pi \cdot \color{blue}{0.5}}, -\sin^{-1} \left(1 - x\right)\right) \]

4. Applied egg-rr 5.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\pi \cdot 0.5}, \sqrt{\pi \cdot 0.5}, -\sin^{-1} \left(1 - x\right)\right)} \]

5. Taylor expanded in x around 0 10.5%

   \[\leadsto \color{blue}{\pi \cdot {\left(\sqrt{0.5}\right)}^{2} - \sin^{-1} \left(1 - x\right)} \]

6. Final simplification 10.5%

   \[\leadsto \pi \cdot {\left(\sqrt{0.5}\right)}^{2} - \sin^{-1} \left(1 - x\right) \]
7. Add Preprocessing

Alternative 4: 6.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \left(1 + \log \left(e^{\cos^{-1} \left(1 - x\right)}\right)\right) + -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (+ 1.0 (log (exp (acos (- 1.0 x))))) -1.0))

C

double code(double x) {
	return (1.0 + log(exp(acos((1.0 - x))))) + -1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 + log(exp(acos((1.0d0 - x))))) + (-1.0d0)
end function

Java

public static double code(double x) {
	return (1.0 + Math.log(Math.exp(Math.acos((1.0 - x))))) + -1.0;
}

Python

def code(x):
	return (1.0 + math.log(math.exp(math.acos((1.0 - x))))) + -1.0

Julia

function code(x)
	return Float64(Float64(1.0 + log(exp(acos(Float64(1.0 - x))))) + -1.0)
end

MATLAB

function tmp = code(x)
	tmp = (1.0 + log(exp(acos((1.0 - x))))) + -1.0;
end

Wolfram

code[x_] := N[(N[(1.0 + N[Log[N[Exp[N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 7.3%

   \[\cos^{-1} \left(1 - x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. expm1-log1p-u 7.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)} \]

   2. expm1-udef 7.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1} \]

   3. log1p-udef 7.3%

      \[\leadsto e^{\color{blue}{\log \left(1 + \cos^{-1} \left(1 - x\right)\right)}} - 1 \]

   4. rem-exp-log 7.3%

      \[\leadsto \color{blue}{\left(1 + \cos^{-1} \left(1 - x\right)\right)} - 1 \]

4. Applied egg-rr 7.3%

   \[\leadsto \color{blue}{\left(1 + \cos^{-1} \left(1 - x\right)\right) - 1} \]
5. Step-by-step derivation

   1. add-log-exp 7.3%

      \[\leadsto \color{blue}{\log \left(e^{\cos^{-1} \left(1 - x\right)}\right)} \]

6. Applied egg-rr 7.3%

   \[\leadsto \left(1 + \color{blue}{\log \left(e^{\cos^{-1} \left(1 - x\right)}\right)}\right) - 1 \]

7. Final simplification 7.3%

   \[\leadsto \left(1 + \log \left(e^{\cos^{-1} \left(1 - x\right)}\right)\right) + -1 \]
8. Add Preprocessing

Alternative 5: 6.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \log \left(e^{\cos^{-1} \left(1 - x\right)}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log (exp (acos (- 1.0 x)))))

C

double code(double x) {
	return log(exp(acos((1.0 - x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log(exp(acos((1.0d0 - x))))
end function

Java

public static double code(double x) {
	return Math.log(Math.exp(Math.acos((1.0 - x))));
}

Python

def code(x):
	return math.log(math.exp(math.acos((1.0 - x))))

Julia

function code(x)
	return log(exp(acos(Float64(1.0 - x))))
end

MATLAB

function tmp = code(x)
	tmp = log(exp(acos((1.0 - x))));
end

Wolfram

code[x_] := N[Log[N[Exp[N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 7.3%

   \[\cos^{-1} \left(1 - x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-log-exp 7.3%

      \[\leadsto \color{blue}{\log \left(e^{\cos^{-1} \left(1 - x\right)}\right)} \]

4. Applied egg-rr 7.3%

   \[\leadsto \color{blue}{\log \left(e^{\cos^{-1} \left(1 - x\right)}\right)} \]

5. Final simplification 7.3%

   \[\leadsto \log \left(e^{\cos^{-1} \left(1 - x\right)}\right) \]
6. Add Preprocessing

Alternative 6: 6.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (acos (- 1.0 x)))

C

double code(double x) {
	return acos((1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = acos((1.0d0 - x))
end function

Java

public static double code(double x) {
	return Math.acos((1.0 - x));
}

Python

def code(x):
	return math.acos((1.0 - x))

Julia

function code(x)
	return acos(Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = acos((1.0 - x));
end

Wolfram

code[x_] := N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 7.3%

   \[\cos^{-1} \left(1 - x\right) \]
2. Add Preprocessing

3. Final simplification 7.3%

   \[\leadsto \cos^{-1} \left(1 - x\right) \]
4. Add Preprocessing

Developer target: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \sin^{-1} \left(\sqrt{\frac{x}{2}}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 2.0 (asin (sqrt (/ x 2.0)))))

C

double code(double x) {
	return 2.0 * asin(sqrt((x / 2.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 * asin(sqrt((x / 2.0d0)))
end function

Java

public static double code(double x) {
	return 2.0 * Math.asin(Math.sqrt((x / 2.0)));
}

Python

def code(x):
	return 2.0 * math.asin(math.sqrt((x / 2.0)))

Julia

function code(x)
	return Float64(2.0 * asin(sqrt(Float64(x / 2.0))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * asin(sqrt((x / 2.0)));
end

Wolfram

code[x_] := N[(2.0 * N[ArcSin[N[Sqrt[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024033 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

  :herbie-target
  (* 2.0 (asin (sqrt (/ x 2.0))))

  (acos (- 1.0 x)))