Given's Rotation SVD example, simplified

Percentage Accurate: 75.4% → 99.9%

Time: 4.1s

Alternatives: 9

Speedup: 2.1×

Specification

Math

\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))

C

double code(double x) {
	return 1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x)))));
}

Java

public static double code(double x) {
	return 1.0 - Math.sqrt((0.5 * (1.0 + (1.0 / Math.hypot(1.0, x)))));
}

Python

def code(x):
	return 1.0 - math.sqrt((0.5 * (1.0 + (1.0 / math.hypot(1.0, x)))))

Julia

function code(x)
	return Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x))))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x)))));
end

Wolfram

code[x_] := N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	(1) - (sqrt(((5e-1) * ((1) + ((1) / (sqrt((((1) ^ (2)) + (x ^ (2))))))))))
END code

Initial Program: 75.4% accurate, 1.0× speedup

Math

\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))

C

double code(double x) {
	return 1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x)))));
}

Java

public static double code(double x) {
	return 1.0 - Math.sqrt((0.5 * (1.0 + (1.0 / Math.hypot(1.0, x)))));
}

Python

def code(x):
	return 1.0 - math.sqrt((0.5 * (1.0 + (1.0 / math.hypot(1.0, x)))))

Julia

function code(x)
	return Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x))))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x)))));
end

Wolfram

code[x_] := N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	(1) - (sqrt(((5e-1) * ((1) + ((1) / (sqrt((((1) ^ (2)) + (x ^ (2))))))))))
END code

Alternative 1: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{0.5}{\sqrt{\mathsf{fma}\left(\left|x\right|, \left|x\right|, 1\right)}}\\ t_1 := {\left(\left|x\right|\right)}^{2}\\ \mathbf{if}\;\left|x\right| \leq 0.0018:\\ \;\;\;\;t\_1 \cdot \left(0.125 + -0.0859375 \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sqrt{t\_0 - -0.5} - -1} \cdot \left(t\_0 - 0.5\right)\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (/ 0.5 (sqrt (fma (fabs x) (fabs x) 1.0))))
       (t_1 (pow (fabs x) 2.0)))
  (if (<= (fabs x) 0.0018)
    (* t_1 (+ 0.125 (* -0.0859375 t_1)))
    (* (/ -1.0 (- (sqrt (- t_0 -0.5)) -1.0)) (- t_0 0.5)))))

C

double code(double x) {
	double t_0 = 0.5 / sqrt(fma(fabs(x), fabs(x), 1.0));
	double t_1 = pow(fabs(x), 2.0);
	double tmp;
	if (fabs(x) <= 0.0018) {
		tmp = t_1 * (0.125 + (-0.0859375 * t_1));
	} else {
		tmp = (-1.0 / (sqrt((t_0 - -0.5)) - -1.0)) * (t_0 - 0.5);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 / sqrt(fma(abs(x), abs(x), 1.0)))
	t_1 = abs(x) ^ 2.0
	tmp = 0.0
	if (abs(x) <= 0.0018)
		tmp = Float64(t_1 * Float64(0.125 + Float64(-0.0859375 * t_1)));
	else
		tmp = Float64(Float64(-1.0 / Float64(sqrt(Float64(t_0 - -0.5)) - -1.0)) * Float64(t_0 - 0.5));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Abs[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.0018], N[(t$95$1 * N[(0.125 + N[(-0.0859375 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[(N[Sqrt[N[(t$95$0 - -0.5), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision]]]]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET t_0 = ((5e-1) / (sqrt((((abs(x)) * (abs(x))) + (1))))) IN
		LET t_1 = ((abs(x)) ^ (2)) IN
			LET tmp = IF ((abs(x)) <= (17999999999999999507338532822586785187013447284698486328125e-61)) THEN (t_1 * ((125e-3) + ((-859375e-7) * t_1))) ELSE (((-1) / ((sqrt((t_0 - (-5e-1)))) - (-1))) * (t_0 - (5e-1))) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if x < 0.0018

   1. Initial program 75.4%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto {x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 49.9%

      \[\leadsto {x}^{2} \cdot \left(0.125 + -0.0859375 \cdot {x}^{2}\right) \]

   if 0.0018 < x

   1. Initial program 75.4%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

   3. Applied rewrites 76.2%

      \[\leadsto \frac{1 - \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5\right)}{1 + \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5}} \]
   4. Step-by-step derivation

   5. Applied rewrites 76.2%

      \[\leadsto \frac{-1}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1} \cdot \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - 0.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{0.5}{\sqrt{\mathsf{fma}\left(\left|x\right|, \left|x\right|, 1\right)}}\\ \mathbf{if}\;\left|x\right| \leq 1.2:\\ \;\;\;\;\left(\left|x\right| \cdot \left|x\right|\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sqrt{t\_0 - -0.5} - -1} \cdot \left(t\_0 - 0.5\right)\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (/ 0.5 (sqrt (fma (fabs x) (fabs x) 1.0)))))
  (if (<= (fabs x) 1.2)
    (* (* (fabs x) (fabs x)) 0.125)
    (* (/ -1.0 (- (sqrt (- t_0 -0.5)) -1.0)) (- t_0 0.5)))))

C

double code(double x) {
	double t_0 = 0.5 / sqrt(fma(fabs(x), fabs(x), 1.0));
	double tmp;
	if (fabs(x) <= 1.2) {
		tmp = (fabs(x) * fabs(x)) * 0.125;
	} else {
		tmp = (-1.0 / (sqrt((t_0 - -0.5)) - -1.0)) * (t_0 - 0.5);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 / sqrt(fma(abs(x), abs(x), 1.0)))
	tmp = 0.0
	if (abs(x) <= 1.2)
		tmp = Float64(Float64(abs(x) * abs(x)) * 0.125);
	else
		tmp = Float64(Float64(-1.0 / Float64(sqrt(Float64(t_0 - -0.5)) - -1.0)) * Float64(t_0 - 0.5));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 1.2], N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision], N[(N[(-1.0 / N[(N[Sqrt[N[(t$95$0 - -0.5), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision]]]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET t_0 = ((5e-1) / (sqrt((((abs(x)) * (abs(x))) + (1))))) IN
		LET tmp = IF ((abs(x)) <= (11999999999999999555910790149937383830547332763671875e-52)) THEN (((abs(x)) * (abs(x))) * (125e-3)) ELSE (((-1) / ((sqrt((t_0 - (-5e-1)))) - (-1))) * (t_0 - (5e-1))) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if x < 1.2

   1. Initial program 75.4%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

   3. Applied rewrites 76.2%

      \[\leadsto \frac{1 - \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5\right)}{1 + \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{1}{8} \cdot {x}^{2} \]
   5. Step-by-step derivation

   6. Applied rewrites 51.4%

      \[\leadsto 0.125 \cdot {x}^{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 51.4%

      \[\leadsto \left(x \cdot x\right) \cdot 0.125 \]

   if 1.2 < x

   1. Initial program 75.4%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

   3. Applied rewrites 76.2%

      \[\leadsto \frac{1 - \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5\right)}{1 + \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5}} \]
   4. Step-by-step derivation

   5. Applied rewrites 76.2%

      \[\leadsto \frac{-1}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1} \cdot \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - 0.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{0.5}{\sqrt{\mathsf{fma}\left(\left|x\right|, \left|x\right|, 1\right)}} - -0.5\\ \mathbf{if}\;\left|x\right| \leq 1.2:\\ \;\;\;\;\left(\left|x\right| \cdot \left|x\right|\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{1 + \sqrt{t\_0}}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (- (/ 0.5 (sqrt (fma (fabs x) (fabs x) 1.0))) -0.5)))
  (if (<= (fabs x) 1.2)
    (* (* (fabs x) (fabs x)) 0.125)
    (/ (- 1.0 t_0) (+ 1.0 (sqrt t_0))))))

C

double code(double x) {
	double t_0 = (0.5 / sqrt(fma(fabs(x), fabs(x), 1.0))) - -0.5;
	double tmp;
	if (fabs(x) <= 1.2) {
		tmp = (fabs(x) * fabs(x)) * 0.125;
	} else {
		tmp = (1.0 - t_0) / (1.0 + sqrt(t_0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(0.5 / sqrt(fma(abs(x), abs(x), 1.0))) - -0.5)
	tmp = 0.0
	if (abs(x) <= 1.2)
		tmp = Float64(Float64(abs(x) * abs(x)) * 0.125);
	else
		tmp = Float64(Float64(1.0 - t_0) / Float64(1.0 + sqrt(t_0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(0.5 / N[Sqrt[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 1.2], N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET t_0 = (((5e-1) / (sqrt((((abs(x)) * (abs(x))) + (1))))) - (-5e-1)) IN
		LET tmp = IF ((abs(x)) <= (11999999999999999555910790149937383830547332763671875e-52)) THEN (((abs(x)) * (abs(x))) * (125e-3)) ELSE (((1) - t_0) / ((1) + (sqrt(t_0)))) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if x < 1.2

   1. Initial program 75.4%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

   3. Applied rewrites 76.2%

      \[\leadsto \frac{1 - \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5\right)}{1 + \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{1}{8} \cdot {x}^{2} \]
   5. Step-by-step derivation

   6. Applied rewrites 51.4%

      \[\leadsto 0.125 \cdot {x}^{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 51.4%

      \[\leadsto \left(x \cdot x\right) \cdot 0.125 \]

   if 1.2 < x

   1. Initial program 75.4%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

   3. Applied rewrites 76.2%

      \[\leadsto \frac{1 - \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5\right)}{1 + \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{0.5}{\left|x\right|} - -0.5\\ \mathbf{if}\;\left|x\right| \leq 2.2:\\ \;\;\;\;\left(\left|x\right| \cdot \left|x\right|\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{1 + \sqrt{t\_0}}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (- (/ 0.5 (fabs x)) -0.5)))
  (if (<= (fabs x) 2.2)
    (* (* (fabs x) (fabs x)) 0.125)
    (/ (- 1.0 t_0) (+ 1.0 (sqrt t_0))))))

C

double code(double x) {
	double t_0 = (0.5 / fabs(x)) - -0.5;
	double tmp;
	if (fabs(x) <= 2.2) {
		tmp = (fabs(x) * fabs(x)) * 0.125;
	} else {
		tmp = (1.0 - t_0) / (1.0 + sqrt(t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 / abs(x)) - (-0.5d0)
    if (abs(x) <= 2.2d0) then
        tmp = (abs(x) * abs(x)) * 0.125d0
    else
        tmp = (1.0d0 - t_0) / (1.0d0 + sqrt(t_0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (0.5 / Math.abs(x)) - -0.5;
	double tmp;
	if (Math.abs(x) <= 2.2) {
		tmp = (Math.abs(x) * Math.abs(x)) * 0.125;
	} else {
		tmp = (1.0 - t_0) / (1.0 + Math.sqrt(t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = (0.5 / math.fabs(x)) - -0.5
	tmp = 0
	if math.fabs(x) <= 2.2:
		tmp = (math.fabs(x) * math.fabs(x)) * 0.125
	else:
		tmp = (1.0 - t_0) / (1.0 + math.sqrt(t_0))
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(0.5 / abs(x)) - -0.5)
	tmp = 0.0
	if (abs(x) <= 2.2)
		tmp = Float64(Float64(abs(x) * abs(x)) * 0.125);
	else
		tmp = Float64(Float64(1.0 - t_0) / Float64(1.0 + sqrt(t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (0.5 / abs(x)) - -0.5;
	tmp = 0.0;
	if (abs(x) <= 2.2)
		tmp = (abs(x) * abs(x)) * 0.125;
	else
		tmp = (1.0 - t_0) / (1.0 + sqrt(t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(0.5 / N[Abs[x], $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 2.2], N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET t_0 = (((5e-1) / (abs(x))) - (-5e-1)) IN
		LET tmp = IF ((abs(x)) <= (220000000000000017763568394002504646778106689453125e-50)) THEN (((abs(x)) * (abs(x))) * (125e-3)) ELSE (((1) - t_0) / ((1) + (sqrt(t_0)))) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 75.4%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

   3. Applied rewrites 76.2%

      \[\leadsto \frac{1 - \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5\right)}{1 + \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{1}{8} \cdot {x}^{2} \]
   5. Step-by-step derivation

   6. Applied rewrites 51.4%

      \[\leadsto 0.125 \cdot {x}^{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 51.4%

      \[\leadsto \left(x \cdot x\right) \cdot 0.125 \]

   if 2.2000000000000002 < x

   1. Initial program 75.4%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

   3. Applied rewrites 76.2%

      \[\leadsto \frac{1 - \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5\right)}{1 + \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{1 - \left(\frac{\frac{1}{2}}{x} - -0.5\right)}{1 + \sqrt{\frac{\frac{1}{2}}{x} - -0.5}} \]
   5. Step-by-step derivation

   6. Applied rewrites 50.0%

      \[\leadsto \frac{1 - \left(\frac{0.5}{x} - -0.5\right)}{1 + \sqrt{\frac{0.5}{x} - -0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.2:\\ \;\;\;\;\left(\left|x\right| \cdot \left|x\right|\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(\left|x\right|, \left|x\right|, 1\right)}} - -0.5}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (if (<= (fabs x) 1.2)
  (* (* (fabs x) (fabs x)) 0.125)
  (- 1.0 (sqrt (- (/ 0.5 (sqrt (fma (fabs x) (fabs x) 1.0))) -0.5)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 1.2) {
		tmp = (fabs(x) * fabs(x)) * 0.125;
	} else {
		tmp = 1.0 - sqrt(((0.5 / sqrt(fma(fabs(x), fabs(x), 1.0))) - -0.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 1.2)
		tmp = Float64(Float64(abs(x) * abs(x)) * 0.125);
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / sqrt(fma(abs(x), abs(x), 1.0))) - -0.5)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 1.2], N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(0.5 / N[Sqrt[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET tmp = IF ((abs(x)) <= (11999999999999999555910790149937383830547332763671875e-52)) THEN (((abs(x)) * (abs(x))) * (125e-3)) ELSE ((1) - (sqrt((((5e-1) / (sqrt((((abs(x)) * (abs(x))) + (1))))) - (-5e-1))))) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if x < 1.2

   1. Initial program 75.4%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

   3. Applied rewrites 76.2%

      \[\leadsto \frac{1 - \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5\right)}{1 + \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{1}{8} \cdot {x}^{2} \]
   5. Step-by-step derivation

   6. Applied rewrites 51.4%

      \[\leadsto 0.125 \cdot {x}^{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 51.4%

      \[\leadsto \left(x \cdot x\right) \cdot 0.125 \]

   if 1.2 < x

   1. Initial program 75.4%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

   3. Applied rewrites 75.5%

      \[\leadsto 1 - \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.25:\\ \;\;\;\;\left(\left|x\right| \cdot \left|x\right|\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (if (<= (fabs x) 1.25)
  (* (* (fabs x) (fabs x)) 0.125)
  (/ 0.5 (+ 1.0 (sqrt 0.5)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 1.25) {
		tmp = (fabs(x) * fabs(x)) * 0.125;
	} else {
		tmp = 0.5 / (1.0 + sqrt(0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (abs(x) <= 1.25d0) then
        tmp = (abs(x) * abs(x)) * 0.125d0
    else
        tmp = 0.5d0 / (1.0d0 + sqrt(0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 1.25) {
		tmp = (Math.abs(x) * Math.abs(x)) * 0.125;
	} else {
		tmp = 0.5 / (1.0 + Math.sqrt(0.5));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 1.25:
		tmp = (math.fabs(x) * math.fabs(x)) * 0.125
	else:
		tmp = 0.5 / (1.0 + math.sqrt(0.5))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 1.25)
		tmp = Float64(Float64(abs(x) * abs(x)) * 0.125);
	else
		tmp = Float64(0.5 / Float64(1.0 + sqrt(0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 1.25)
		tmp = (abs(x) * abs(x)) * 0.125;
	else
		tmp = 0.5 / (1.0 + sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 1.25], N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision], N[(0.5 / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET tmp = IF ((abs(x)) <= (125e-2)) THEN (((abs(x)) * (abs(x))) * (125e-3)) ELSE ((5e-1) / ((1) + (sqrt((5e-1))))) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 75.4%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

   3. Applied rewrites 76.2%

      \[\leadsto \frac{1 - \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5\right)}{1 + \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{1}{8} \cdot {x}^{2} \]
   5. Step-by-step derivation

   6. Applied rewrites 51.4%

      \[\leadsto 0.125 \cdot {x}^{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 51.4%

      \[\leadsto \left(x \cdot x\right) \cdot 0.125 \]

   if 1.25 < x

   1. Initial program 75.4%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

   3. Applied rewrites 76.2%

      \[\leadsto \frac{1 - \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5\right)}{1 + \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{2}}{1 + \sqrt{\frac{1}{2}}} \]
   5. Step-by-step derivation

   6. Applied rewrites 51.3%

      \[\leadsto \frac{0.5}{1 + \sqrt{0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 97.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.25:\\ \;\;\;\;\left(\left|x\right| \cdot \left|x\right|\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;0.2928932188134524\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (if (<= (fabs x) 1.25)
  (* (* (fabs x) (fabs x)) 0.125)
  0.2928932188134524))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 1.25) {
		tmp = (fabs(x) * fabs(x)) * 0.125;
	} else {
		tmp = 0.2928932188134524;
	}
	return tmp;
}

Fortran

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (abs(x) <= 1.25d0) then
        tmp = (abs(x) * abs(x)) * 0.125d0
    else
        tmp = 0.2928932188134524d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 1.25) {
		tmp = (Math.abs(x) * Math.abs(x)) * 0.125;
	} else {
		tmp = 0.2928932188134524;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 1.25:
		tmp = (math.fabs(x) * math.fabs(x)) * 0.125
	else:
		tmp = 0.2928932188134524
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 1.25)
		tmp = Float64(Float64(abs(x) * abs(x)) * 0.125);
	else
		tmp = 0.2928932188134524;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 1.25)
		tmp = (abs(x) * abs(x)) * 0.125;
	else
		tmp = 0.2928932188134524;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 1.25], N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision], 0.2928932188134524]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET tmp = IF ((abs(x)) <= (125e-2)) THEN (((abs(x)) * (abs(x))) * (125e-3)) ELSE (29289321881345242726268907063058577477931976318359375e-53) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 75.4%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

   3. Applied rewrites 76.2%

      \[\leadsto \frac{1 - \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5\right)}{1 + \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{1}{8} \cdot {x}^{2} \]
   5. Step-by-step derivation

   6. Applied rewrites 51.4%

      \[\leadsto 0.125 \cdot {x}^{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 51.4%

      \[\leadsto \left(x \cdot x\right) \cdot 0.125 \]

   if 1.25 < x

   1. Initial program 75.4%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

   3. Applied rewrites 51.5%

      \[\leadsto 1 - \sqrt{\frac{1}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1} \cdot \sqrt{0.5} \]

   4. Evaluated real constant 51.5%

      \[\leadsto 1 - \sqrt{\frac{1}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1} \cdot 0.7071067811865476 \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{2638147582215219}{9007199254740992} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.5%

      \[\leadsto 0.2928932188134524 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 74.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5.5 \cdot 10^{-79}:\\ \;\;\;\;1 - \sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;0.2928932188134524\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (if (<= (fabs x) 5.5e-79) (- 1.0 (sqrt 1.0)) 0.2928932188134524))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 5.5e-79) {
		tmp = 1.0 - sqrt(1.0);
	} else {
		tmp = 0.2928932188134524;
	}
	return tmp;
}

Fortran

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (abs(x) <= 5.5d-79) then
        tmp = 1.0d0 - sqrt(1.0d0)
    else
        tmp = 0.2928932188134524d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 5.5e-79) {
		tmp = 1.0 - Math.sqrt(1.0);
	} else {
		tmp = 0.2928932188134524;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 5.5e-79:
		tmp = 1.0 - math.sqrt(1.0)
	else:
		tmp = 0.2928932188134524
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 5.5e-79)
		tmp = Float64(1.0 - sqrt(1.0));
	else
		tmp = 0.2928932188134524;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 5.5e-79)
		tmp = 1.0 - sqrt(1.0);
	else
		tmp = 0.2928932188134524;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 5.5e-79], N[(1.0 - N[Sqrt[1.0], $MachinePrecision]), $MachinePrecision], 0.2928932188134524]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET tmp = IF ((abs(x)) <= (54999999999999996942057192297876001245053568316894463337206637269566867377234666277132519617507187437460990248639054916944318873802304893281316808212413385913220564905970669705645035915602216770869947737310212687589228153228759765625e-311)) THEN ((1) - (sqrt((1)))) ELSE (29289321881345242726268907063058577477931976318359375e-53) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if x < 5.4999999999999997e-79

   1. Initial program 75.4%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]

   2. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\frac{1}{2}} \]
   3. Step-by-step derivation

   4. Applied rewrites 50.5%

      \[\leadsto 1 - \sqrt{0.5} \]

   5. Taylor expanded in x around 0

      \[\leadsto 1 - \sqrt{1} \]
   6. Step-by-step derivation

   7. Applied rewrites 26.9%

      \[\leadsto 1 - \sqrt{1} \]

   if 5.4999999999999997e-79 < x

   1. Initial program 75.4%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

   3. Applied rewrites 51.5%

      \[\leadsto 1 - \sqrt{\frac{1}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1} \cdot \sqrt{0.5} \]

   4. Evaluated real constant 51.5%

      \[\leadsto 1 - \sqrt{\frac{1}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1} \cdot 0.7071067811865476 \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{2638147582215219}{9007199254740992} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.5%

      \[\leadsto 0.2928932188134524 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 50.5% accurate, 29.6× speedup

Math

\[0.2928932188134524 \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  0.2928932188134524)

C

double code(double x) {
	return 0.2928932188134524;
}

Fortran

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = 0.2928932188134524d0
end function

Java

public static double code(double x) {
	return 0.2928932188134524;
}

Python

def code(x):
	return 0.2928932188134524

Julia

function code(x)
	return 0.2928932188134524
end

MATLAB

function tmp = code(x)
	tmp = 0.2928932188134524;
end

Wolfram

code[x_] := 0.2928932188134524

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	29289321881345242726268907063058577477931976318359375e-53
END code

Derivation

1. Initial program 75.4%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
2. Step-by-step derivation

3. Applied rewrites 51.5%

   \[\leadsto 1 - \sqrt{\frac{1}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1} \cdot \sqrt{0.5} \]

4. Evaluated real constant 51.5%

   \[\leadsto 1 - \sqrt{\frac{1}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1} \cdot 0.7071067811865476 \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{2638147582215219}{9007199254740992} \]
6. Step-by-step derivation

7. Applied rewrites 50.5%

   \[\leadsto 0.2928932188134524 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025355 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))