Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.1% → 99.1%

Time: 13.3s

Alternatives: 8

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))

C

double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = acos(((1.0d0 - (5.0d0 * (v * v))) / ((v * v) - 1.0d0)))
end function

Java

public static double code(double v) {
	return Math.acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

Python

def code(v):
	return math.acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)))

Julia

function code(v)
	return acos(Float64(Float64(1.0 - Float64(5.0 * Float64(v * v))) / Float64(Float64(v * v) - 1.0)))
end

MATLAB

function tmp = code(v)
	tmp = acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
end

Wolfram

code[v_] := N[ArcCos[N[(N[(1.0 - N[(5.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(v * v), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))

C

double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = acos(((1.0d0 - (5.0d0 * (v * v))) / ((v * v) - 1.0d0)))
end function

Java

public static double code(double v) {
	return Math.acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

Python

def code(v):
	return math.acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)))

Julia

function code(v)
	return acos(Float64(Float64(1.0 - Float64(5.0 * Float64(v * v))) / Float64(Float64(v * v) - 1.0)))
end

MATLAB

function tmp = code(v)
	tmp = acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
end

Wolfram

code[v_] := N[ArcCos[N[(N[(1.0 - N[(5.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(v * v), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{PI}\left(\right) - \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot 5, v, -1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (- (PI) (acos (/ (fma (* v 5.0) v -1.0) (fma v v -1.0)))))

Derivation

1. Initial program 99.3%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Add Preprocessing

4. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\mathsf{PI}\left(\right) - \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot 5, v, -1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (acos (/ (fma -5.0 (* v v) 1.0) (fma v v -1.0))))

C

double code(double v) {
	return acos((fma(-5.0, (v * v), 1.0) / fma(v, v, -1.0)));
}

Julia

function code(v)
	return acos(Float64(fma(-5.0, Float64(v * v), 1.0) / fma(v, v, -1.0)))
end

Wolfram

code[v_] := N[ArcCos[N[(N[(-5.0 * N[(v * v), $MachinePrecision] + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{1 - 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{1 - \color{blue}{5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right) \]

   3. fp-cancel-sub-sign-inv N/A

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{1 + \left(\mathsf{neg}\left(5\right)\right) \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right) \]

   4. +-commutative N/A

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\left(\mathsf{neg}\left(5\right)\right) \cdot \left(v \cdot v\right) + 1}}{v \cdot v - 1}\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(5\right), v \cdot v, 1\right)}}{v \cdot v - 1}\right) \]

   6. metadata-eval 99.3

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(\color{blue}{-5}, v \cdot v, 1\right)}{v \cdot v - 1}\right) \]

   7. lift--.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\color{blue}{v \cdot v - 1}}\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\color{blue}{v \cdot v} - 1}\right) \]

   9. difference-of-sqr-1 N/A

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\color{blue}{\left(v + 1\right) \cdot \left(v - 1\right)}}\right) \]

   10. difference-of-sqr--1-rev N/A

       \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\color{blue}{v \cdot v + -1}}\right) \]

   11. lower-fma.f64 99.3

       \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\color{blue}{\mathsf{fma}\left(v, v, -1\right)}}\right) \]

4. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(-\mathsf{fma}\left(\left(-4 \cdot v\right) \cdot v - 4, v \cdot v, 1\right)\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (acos (- (fma (- (* (* -4.0 v) v) 4.0) (* v v) 1.0))))

C

double code(double v) {
	return acos(-fma((((-4.0 * v) * v) - 4.0), (v * v), 1.0));
}

Julia

function code(v)
	return acos(Float64(-fma(Float64(Float64(Float64(-4.0 * v) * v) - 4.0), Float64(v * v), 1.0)))
end

Wolfram

code[v_] := N[ArcCos[(-N[(N[(N[(N[(-4.0 * v), $MachinePrecision] * v), $MachinePrecision] - 4.0), $MachinePrecision] * N[(v * v), $MachinePrecision] + 1.0), $MachinePrecision])], $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Add Preprocessing

4. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\mathsf{PI}\left(\right) - \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot 5, v, -1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \]

5. Taylor expanded in v around 0

   \[\leadsto \mathsf{PI}\left(\right) - \cos^{-1} \color{blue}{\left(1 + {v}^{2} \cdot \left({v}^{2} \cdot \left(-4 \cdot {v}^{2} - 4\right) - 4\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{PI}\left(\right) - \cos^{-1} \color{blue}{\left({v}^{2} \cdot \left({v}^{2} \cdot \left(-4 \cdot {v}^{2} - 4\right) - 4\right) + 1\right)} \]

   2. *-commutative N/A

      \[\leadsto \mathsf{PI}\left(\right) - \cos^{-1} \left(\color{blue}{\left({v}^{2} \cdot \left(-4 \cdot {v}^{2} - 4\right) - 4\right) \cdot {v}^{2}} + 1\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{PI}\left(\right) - \cos^{-1} \color{blue}{\left(\mathsf{fma}\left({v}^{2} \cdot \left(-4 \cdot {v}^{2} - 4\right) - 4, {v}^{2}, 1\right)\right)} \]

7. Applied rewrites 99.2%

   \[\leadsto \mathsf{PI}\left(\right) - \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\left(-4 \cdot \left(v \cdot v\right) - 4\right) \cdot \left(v \cdot v\right) - 4, v \cdot v, 1\right)\right)} \]
8. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\mathsf{PI}\left(\right) - \cos^{-1} \left(\mathsf{fma}\left(\left(-4 \cdot \left(v \cdot v\right) - 4\right) \cdot \left(v \cdot v\right) - 4, v \cdot v, 1\right)\right)} \]

   2. lift-PI.f64 N/A

      \[\leadsto \color{blue}{\mathsf{PI}\left(\right)} - \cos^{-1} \left(\mathsf{fma}\left(\left(-4 \cdot \left(v \cdot v\right) - 4\right) \cdot \left(v \cdot v\right) - 4, v \cdot v, 1\right)\right) \]

   3. lift-acos.f64 N/A

      \[\leadsto \mathsf{PI}\left(\right) - \color{blue}{\cos^{-1} \left(\mathsf{fma}\left(\left(-4 \cdot \left(v \cdot v\right) - 4\right) \cdot \left(v \cdot v\right) - 4, v \cdot v, 1\right)\right)} \]

   4. acos-neg-rev N/A

      \[\leadsto \color{blue}{\cos^{-1} \left(\mathsf{neg}\left(\mathsf{fma}\left(\left(-4 \cdot \left(v \cdot v\right) - 4\right) \cdot \left(v \cdot v\right) - 4, v \cdot v, 1\right)\right)\right)} \]

   5. lower-acos.f64 N/A

      \[\leadsto \color{blue}{\cos^{-1} \left(\mathsf{neg}\left(\mathsf{fma}\left(\left(-4 \cdot \left(v \cdot v\right) - 4\right) \cdot \left(v \cdot v\right) - 4, v \cdot v, 1\right)\right)\right)} \]

9. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\cos^{-1} \left(-\mathsf{fma}\left(\left(\left(-4 \cdot \left(v \cdot v\right) - 4\right) \cdot v\right) \cdot v - 4, v \cdot v, 1\right)\right)} \]

10. Taylor expanded in v around 0

    \[\leadsto \cos^{-1} \left(-\mathsf{fma}\left(\left(-4 \cdot v\right) \cdot v - 4, v \cdot v, 1\right)\right) \]
11. Step-by-step derivation

12. Applied rewrites 99.2%

    \[\leadsto \cos^{-1} \left(-\mathsf{fma}\left(\left(-4 \cdot v\right) \cdot v - 4, v \cdot v, 1\right)\right) \]
13. Add Preprocessing

Alternative 4: 98.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{PI}\left(\right) - \cos^{-1} \left(\mathsf{fma}\left(-4, v \cdot v, 1\right)\right) \end{array} \]

FPCore

(FPCore (v) :precision binary64 (- (PI) (acos (fma -4.0 (* v v) 1.0))))

Derivation

1. Initial program 99.3%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Add Preprocessing

4. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\mathsf{PI}\left(\right) - \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot 5, v, -1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \]

5. Taylor expanded in v around 0

   \[\leadsto \mathsf{PI}\left(\right) - \cos^{-1} \color{blue}{\left(1 + -4 \cdot {v}^{2}\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{PI}\left(\right) - \cos^{-1} \color{blue}{\left(-4 \cdot {v}^{2} + 1\right)} \]

   2. lower-fma.f64 N/A

      \[\leadsto \mathsf{PI}\left(\right) - \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(-4, {v}^{2}, 1\right)\right)} \]

   3. unpow2 N/A

      \[\leadsto \mathsf{PI}\left(\right) - \cos^{-1} \left(\mathsf{fma}\left(-4, \color{blue}{v \cdot v}, 1\right)\right) \]

   4. lower-*.f64 98.9

      \[\leadsto \mathsf{PI}\left(\right) - \cos^{-1} \left(\mathsf{fma}\left(-4, \color{blue}{v \cdot v}, 1\right)\right) \]

7. Applied rewrites 98.9%

   \[\leadsto \mathsf{PI}\left(\right) - \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(-4, v \cdot v, 1\right)\right)} \]
8. Add Preprocessing

Alternative 5: 98.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(4 \cdot \left(v \cdot v\right) - 1\right) \end{array} \]

FPCore

(FPCore (v) :precision binary64 (acos (- (* 4.0 (* v v)) 1.0)))

C

double code(double v) {
	return acos(((4.0 * (v * v)) - 1.0));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = acos(((4.0d0 * (v * v)) - 1.0d0))
end function

Java

public static double code(double v) {
	return Math.acos(((4.0 * (v * v)) - 1.0));
}

Python

def code(v):
	return math.acos(((4.0 * (v * v)) - 1.0))

Julia

function code(v)
	return acos(Float64(Float64(4.0 * Float64(v * v)) - 1.0))
end

MATLAB

function tmp = code(v)
	tmp = acos(((4.0 * (v * v)) - 1.0));
end

Wolfram

code[v_] := N[ArcCos[N[(N[(4.0 * N[(v * v), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \cos^{-1} \color{blue}{\left(4 \cdot {v}^{2} - 1\right)} \]
4. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(4 \cdot {v}^{2} - 1\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{4 \cdot {v}^{2}} - 1\right) \]

   3. unpow2 N/A

      \[\leadsto \cos^{-1} \left(4 \cdot \color{blue}{\left(v \cdot v\right)} - 1\right) \]

   4. lower-*.f64 98.9

      \[\leadsto \cos^{-1} \left(4 \cdot \color{blue}{\left(v \cdot v\right)} - 1\right) \]

5. Applied rewrites 98.9%

   \[\leadsto \cos^{-1} \color{blue}{\left(4 \cdot \left(v \cdot v\right) - 1\right)} \]
6. Add Preprocessing

Alternative 6: 97.8% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (v) :precision binary64 (acos (* (- v 1.0) (+ v 1.0))))

C

double code(double v) {
	return acos(((v - 1.0) * (v + 1.0)));
}

Fortran

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

Java

public static double code(double v) {
	return Math.acos(((v - 1.0) * (v + 1.0)));
}

Python

def code(v):
	return math.acos(((v - 1.0) * (v + 1.0)))

Julia

function code(v)
	return acos(Float64(Float64(v - 1.0) * Float64(v + 1.0)))
end

MATLAB

function tmp = code(v)
	tmp = acos(((v - 1.0) * (v + 1.0)));
end

Wolfram

code[v_] := N[ArcCos[N[(N[(v - 1.0), $MachinePrecision] * N[(v + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Add Preprocessing

4. Applied rewrites 99.3%

   \[\leadsto \cos^{-1} \color{blue}{\left(\frac{\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{v + 1}}{\mathsf{fma}\left(v, v, -1\right)} \cdot \left(v + 1\right)\right)} \]

5. Taylor expanded in v around 0

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

   1. lower--.f64 98.0

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

7. Applied rewrites 98.0%

   \[\leadsto \cos^{-1} \left(\color{blue}{\left(v - 1\right)} \cdot \left(v + 1\right)\right) \]
8. Add Preprocessing

Alternative 7: 97.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(v)
	return acos(-1.0)
end

MATLAB

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

Wolfram

code[v_] := N[ArcCos[-1.0], $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in v around inf

   \[\leadsto \cos^{-1} \color{blue}{-5} \]
4. Step-by-step derivation

5. Applied rewrites 0.0%

   \[\leadsto \cos^{-1} \color{blue}{-5} \]

6. Taylor expanded in v around 0

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

8. Applied rewrites 97.9%

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

Alternative 8: 0.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} -5 \end{array} \]

FPCore

(FPCore (v) :precision binary64 (acos -5.0))

C

double code(double v) {
	return acos(-5.0);
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = acos((-5.0d0))
end function

Java

public static double code(double v) {
	return Math.acos(-5.0);
}

Python

def code(v):
	return math.acos(-5.0)

Julia

function code(v)
	return acos(-5.0)
end

MATLAB

function tmp = code(v)
	tmp = acos(-5.0);
end

Wolfram

code[v_] := N[ArcCos[-5.0], $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in v around inf

   \[\leadsto \cos^{-1} \color{blue}{-5} \]
4. Step-by-step derivation

5. Applied rewrites 0.0%

   \[\leadsto \cos^{-1} \color{blue}{-5} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024338 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))