Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 20.2s

Alternatives: 8

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))

C

double code(double v) {
	return ((sqrt(2.0) / 4.0) * sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = ((sqrt(2.0d0) / 4.0d0) * sqrt((1.0d0 - (3.0d0 * (v * v))))) * (1.0d0 - (v * v))
end function

Java

public static double code(double v) {
	return ((Math.sqrt(2.0) / 4.0) * Math.sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v));
}

Python

def code(v):
	return ((math.sqrt(2.0) / 4.0) * math.sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v))

Julia

function code(v)
	return Float64(Float64(Float64(sqrt(2.0) / 4.0) * sqrt(Float64(1.0 - Float64(3.0 * Float64(v * v))))) * Float64(1.0 - Float64(v * v)))
end

MATLAB

function tmp = code(v)
	tmp = ((sqrt(2.0) / 4.0) * sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v));
end

Wolfram

code[v_] := N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / 4.0), $MachinePrecision] * N[Sqrt[N[(1.0 - N[(3.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))

C

double code(double v) {
	return ((sqrt(2.0) / 4.0) * sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = ((sqrt(2.0d0) / 4.0d0) * sqrt((1.0d0 - (3.0d0 * (v * v))))) * (1.0d0 - (v * v))
end function

Java

public static double code(double v) {
	return ((Math.sqrt(2.0) / 4.0) * Math.sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v));
}

Python

def code(v):
	return ((math.sqrt(2.0) / 4.0) * math.sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v))

Julia

function code(v)
	return Float64(Float64(Float64(sqrt(2.0) / 4.0) * sqrt(Float64(1.0 - Float64(3.0 * Float64(v * v))))) * Float64(1.0 - Float64(v * v)))
end

MATLAB

function tmp = code(v)
	tmp = ((sqrt(2.0) / 4.0) * sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v));
end

Wolfram

code[v_] := N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / 4.0), $MachinePrecision] * N[Sqrt[N[(1.0 - N[(3.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{1 + \left(v \cdot v\right) \cdot -3} \cdot \left(\sqrt{2} \cdot \left(1 - v \cdot v\right)\right)}{4} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ (* (sqrt (+ 1.0 (* (* v v) -3.0))) (* (sqrt 2.0) (- 1.0 (* v v)))) 4.0))

C

double code(double v) {
	return (sqrt((1.0 + ((v * v) * -3.0))) * (sqrt(2.0) * (1.0 - (v * v)))) / 4.0;
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = (sqrt((1.0d0 + ((v * v) * (-3.0d0)))) * (sqrt(2.0d0) * (1.0d0 - (v * v)))) / 4.0d0
end function

Java

public static double code(double v) {
	return (Math.sqrt((1.0 + ((v * v) * -3.0))) * (Math.sqrt(2.0) * (1.0 - (v * v)))) / 4.0;
}

Python

def code(v):
	return (math.sqrt((1.0 + ((v * v) * -3.0))) * (math.sqrt(2.0) * (1.0 - (v * v)))) / 4.0

Julia

function code(v)
	return Float64(Float64(sqrt(Float64(1.0 + Float64(Float64(v * v) * -3.0))) * Float64(sqrt(2.0) * Float64(1.0 - Float64(v * v)))) / 4.0)
end

MATLAB

function tmp = code(v)
	tmp = (sqrt((1.0 + ((v * v) * -3.0))) * (sqrt(2.0) * (1.0 - (v * v)))) / 4.0;
end

Wolfram

code[v_] := N[(N[(N[Sqrt[N[(1.0 + N[(N[(v * v), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))

C

double code(double v) {
	return ((sqrt(2.0) / 4.0) * sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = ((sqrt(2.0d0) / 4.0d0) * sqrt((1.0d0 - (3.0d0 * (v * v))))) * (1.0d0 - (v * v))
end function

Java

public static double code(double v) {
	return ((Math.sqrt(2.0) / 4.0) * Math.sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v));
}

Python

def code(v):
	return ((math.sqrt(2.0) / 4.0) * math.sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v))

Julia

function code(v)
	return Float64(Float64(Float64(sqrt(2.0) / 4.0) * sqrt(Float64(1.0 - Float64(3.0 * Float64(v * v))))) * Float64(1.0 - Float64(v * v)))
end

MATLAB

function tmp = code(v)
	tmp = ((sqrt(2.0) / 4.0) * sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v));
end

Wolfram

code[v_] := N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / 4.0), $MachinePrecision] * N[Sqrt[N[(1.0 - N[(3.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 3: 99.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(v \cdot v\right) \cdot -0.625 + \left(0.25 + v \cdot \left(\left(v \cdot \left(v \cdot v\right)\right) \cdot \left(0.09375 + v \cdot \left(v \cdot -0.140625\right)\right)\right)\right)\right) \cdot \sqrt{2} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (*
  (+
   (* (* v v) -0.625)
   (+ 0.25 (* v (* (* v (* v v)) (+ 0.09375 (* v (* v -0.140625)))))))
  (sqrt 2.0)))

C

double code(double v) {
	return (((v * v) * -0.625) + (0.25 + (v * ((v * (v * v)) * (0.09375 + (v * (v * -0.140625))))))) * sqrt(2.0);
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = (((v * v) * (-0.625d0)) + (0.25d0 + (v * ((v * (v * v)) * (0.09375d0 + (v * (v * (-0.140625d0)))))))) * sqrt(2.0d0)
end function

Java

public static double code(double v) {
	return (((v * v) * -0.625) + (0.25 + (v * ((v * (v * v)) * (0.09375 + (v * (v * -0.140625))))))) * Math.sqrt(2.0);
}

Python

def code(v):
	return (((v * v) * -0.625) + (0.25 + (v * ((v * (v * v)) * (0.09375 + (v * (v * -0.140625))))))) * math.sqrt(2.0)

Julia

function code(v)
	return Float64(Float64(Float64(Float64(v * v) * -0.625) + Float64(0.25 + Float64(v * Float64(Float64(v * Float64(v * v)) * Float64(0.09375 + Float64(v * Float64(v * -0.140625))))))) * sqrt(2.0))
end

MATLAB

function tmp = code(v)
	tmp = (((v * v) * -0.625) + (0.25 + (v * ((v * (v * v)) * (0.09375 + (v * (v * -0.140625))))))) * sqrt(2.0);
end

Wolfram

code[v_] := N[(N[(N[(N[(v * v), $MachinePrecision] * -0.625), $MachinePrecision] + N[(0.25 + N[(v * N[(N[(v * N[(v * v), $MachinePrecision]), $MachinePrecision] * N[(0.09375 + N[(v * N[(v * -0.140625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

5. Taylor expanded in v around 0 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]

8. Taylor expanded in v around 0 0

   \[\leadsto expr\]

10. Simplified 0

    \[\leadsto expr\]

12. Applied egg-rr 0

    \[\leadsto expr\]
13. Add Preprocessing

Alternative 4: 99.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{2} \cdot \left(0.25 + \left(v \cdot v\right) \cdot \left(-0.625 + \left(v \cdot v\right) \cdot \left(0.09375 + v \cdot \left(v \cdot -0.140625\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (*
  (sqrt 2.0)
  (+
   0.25
   (* (* v v) (+ -0.625 (* (* v v) (+ 0.09375 (* v (* v -0.140625)))))))))

C

double code(double v) {
	return sqrt(2.0) * (0.25 + ((v * v) * (-0.625 + ((v * v) * (0.09375 + (v * (v * -0.140625)))))));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt(2.0d0) * (0.25d0 + ((v * v) * ((-0.625d0) + ((v * v) * (0.09375d0 + (v * (v * (-0.140625d0))))))))
end function

Java

public static double code(double v) {
	return Math.sqrt(2.0) * (0.25 + ((v * v) * (-0.625 + ((v * v) * (0.09375 + (v * (v * -0.140625)))))));
}

Python

def code(v):
	return math.sqrt(2.0) * (0.25 + ((v * v) * (-0.625 + ((v * v) * (0.09375 + (v * (v * -0.140625)))))))

Julia

function code(v)
	return Float64(sqrt(2.0) * Float64(0.25 + Float64(Float64(v * v) * Float64(-0.625 + Float64(Float64(v * v) * Float64(0.09375 + Float64(v * Float64(v * -0.140625))))))))
end

MATLAB

function tmp = code(v)
	tmp = sqrt(2.0) * (0.25 + ((v * v) * (-0.625 + ((v * v) * (0.09375 + (v * (v * -0.140625)))))));
end

Wolfram

code[v_] := N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.25 + N[(N[(v * v), $MachinePrecision] * N[(-0.625 + N[(N[(v * v), $MachinePrecision] * N[(0.09375 + N[(v * N[(v * -0.140625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

5. Taylor expanded in v around 0 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]

8. Taylor expanded in v around 0 0

   \[\leadsto expr\]

10. Simplified 0

    \[\leadsto expr\]

11. Taylor expanded in v around 0 0

    \[\leadsto expr\]

13. Simplified 0

    \[\leadsto expr\]
14. Add Preprocessing

Alternative 5: 99.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{2} \cdot \left(0.25 + \left(v \cdot v\right) \cdot \left(-0.625 + \left(v \cdot v\right) \cdot 0.09375\right)\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (* (sqrt 2.0) (+ 0.25 (* (* v v) (+ -0.625 (* (* v v) 0.09375))))))

C

double code(double v) {
	return sqrt(2.0) * (0.25 + ((v * v) * (-0.625 + ((v * v) * 0.09375))));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt(2.0d0) * (0.25d0 + ((v * v) * ((-0.625d0) + ((v * v) * 0.09375d0))))
end function

Java

public static double code(double v) {
	return Math.sqrt(2.0) * (0.25 + ((v * v) * (-0.625 + ((v * v) * 0.09375))));
}

Python

def code(v):
	return math.sqrt(2.0) * (0.25 + ((v * v) * (-0.625 + ((v * v) * 0.09375))))

Julia

function code(v)
	return Float64(sqrt(2.0) * Float64(0.25 + Float64(Float64(v * v) * Float64(-0.625 + Float64(Float64(v * v) * 0.09375)))))
end

MATLAB

function tmp = code(v)
	tmp = sqrt(2.0) * (0.25 + ((v * v) * (-0.625 + ((v * v) * 0.09375))));
end

Wolfram

code[v_] := N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.25 + N[(N[(v * v), $MachinePrecision] * N[(-0.625 + N[(N[(v * v), $MachinePrecision] * 0.09375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

5. Taylor expanded in v around 0 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]

8. Taylor expanded in v around 0 0

   \[\leadsto expr\]

10. Simplified 0

    \[\leadsto expr\]
11. Add Preprocessing

Alternative 6: 99.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{2}}{\frac{1}{\left(v \cdot v\right) \cdot -0.625 + 0.25}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ (sqrt 2.0) (/ 1.0 (+ (* (* v v) -0.625) 0.25))))

C

double code(double v) {
	return sqrt(2.0) / (1.0 / (((v * v) * -0.625) + 0.25));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt(2.0d0) / (1.0d0 / (((v * v) * (-0.625d0)) + 0.25d0))
end function

Java

public static double code(double v) {
	return Math.sqrt(2.0) / (1.0 / (((v * v) * -0.625) + 0.25));
}

Python

def code(v):
	return math.sqrt(2.0) / (1.0 / (((v * v) * -0.625) + 0.25))

Julia

function code(v)
	return Float64(sqrt(2.0) / Float64(1.0 / Float64(Float64(Float64(v * v) * -0.625) + 0.25)))
end

MATLAB

function tmp = code(v)
	tmp = sqrt(2.0) / (1.0 / (((v * v) * -0.625) + 0.25));
end

Wolfram

code[v_] := N[(N[Sqrt[2.0], $MachinePrecision] / N[(1.0 / N[(N[(N[(v * v), $MachinePrecision] * -0.625), $MachinePrecision] + 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

5. Taylor expanded in v around 0 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]

8. Taylor expanded in v around 0 0

   \[\leadsto expr\]

10. Simplified 0

    \[\leadsto expr\]

12. Applied egg-rr 0

    \[\leadsto expr\]
13. Add Preprocessing

Alternative 7: 99.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{2} \cdot \left(0.25 + \left(v \cdot v\right) \cdot -0.625\right) \end{array} \]

FPCore

(FPCore (v) :precision binary64 (* (sqrt 2.0) (+ 0.25 (* (* v v) -0.625))))

C

double code(double v) {
	return sqrt(2.0) * (0.25 + ((v * v) * -0.625));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt(2.0d0) * (0.25d0 + ((v * v) * (-0.625d0)))
end function

Java

public static double code(double v) {
	return Math.sqrt(2.0) * (0.25 + ((v * v) * -0.625));
}

Python

def code(v):
	return math.sqrt(2.0) * (0.25 + ((v * v) * -0.625))

Julia

function code(v)
	return Float64(sqrt(2.0) * Float64(0.25 + Float64(Float64(v * v) * -0.625)))
end

MATLAB

function tmp = code(v)
	tmp = sqrt(2.0) * (0.25 + ((v * v) * -0.625));
end

Wolfram

code[v_] := N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.25 + N[(N[(v * v), $MachinePrecision] * -0.625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

5. Taylor expanded in v around 0 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]

8. Taylor expanded in v around 0 0

   \[\leadsto expr\]

10. Simplified 0

    \[\leadsto expr\]
11. Add Preprocessing

Alternative 8: 99.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{2}}{4} \end{array} \]

FPCore

(FPCore (v) :precision binary64 (/ (sqrt 2.0) 4.0))

C

double code(double v) {
	return sqrt(2.0) / 4.0;
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt(2.0d0) / 4.0d0
end function

Java

public static double code(double v) {
	return Math.sqrt(2.0) / 4.0;
}

Python

def code(v):
	return math.sqrt(2.0) / 4.0

Julia

function code(v)
	return Float64(sqrt(2.0) / 4.0)
end

MATLAB

function tmp = code(v)
	tmp = sqrt(2.0) / 4.0;
end

Wolfram

code[v_] := N[(N[Sqrt[2.0], $MachinePrecision] / 4.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

5. Taylor expanded in v around 0 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))