ABCF->ab-angle a

Average Error: 52.4 → 40.2

Time: 20.4s

Precision: binary64

Math

\[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

↓

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := {\left(\frac{t_1}{\left(\sqrt{2 \cdot t_1} \cdot \sqrt{F}\right) \cdot \left(-\sqrt{C + \left(A + t_0\right)}\right)}\right)}^{-1}\\ t_3 := \sqrt{2 \cdot \left(F \cdot t_1\right)}\\ t_4 := \sqrt{t_0}\\ t_5 := \frac{t_3 \cdot \left(-\sqrt{\mathsf{fma}\left(t_4, t_4, A + C\right)}\right)}{t_1}\\ \mathbf{if}\;A \leq -2.2 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{-0.5 \cdot \frac{F}{A}}\right)\\ \mathbf{elif}\;A \leq 7.5 \cdot 10^{-260}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;A \leq 7 \cdot 10^{-126}:\\ \;\;\;\;\frac{t_3 \cdot \left(-\sqrt{t_0 + \left(A + C\right)}\right)}{t_1}\\ \mathbf{elif}\;A \leq 2 \cdot 10^{-74}:\\ \;\;\;\;{\left(-\frac{\sqrt{\frac{C \cdot -2}{F}}}{\sqrt{2}}\right)}^{-1}\\ \mathbf{elif}\;A \leq 5.7 \cdot 10^{-12}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;A \leq 5.5 \cdot 10^{+107}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_5\\ \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (/
  (-
   (sqrt
    (*
     (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F))
     (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
  (- (pow B 2.0) (* (* 4.0 A) C))))

↓

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (hypot B (- A C)))
        (t_1 (fma B B (* A (* C -4.0))))
        (t_2
         (pow
          (/
           t_1
           (* (* (sqrt (* 2.0 t_1)) (sqrt F)) (- (sqrt (+ C (+ A t_0))))))
          -1.0))
        (t_3 (sqrt (* 2.0 (* F t_1))))
        (t_4 (sqrt t_0))
        (t_5 (/ (* t_3 (- (sqrt (fma t_4 t_4 (+ A C))))) t_1)))
   (if (<= A -2.2e+24)
     (* (sqrt 2.0) (- (sqrt (* -0.5 (/ F A)))))
     (if (<= A 7.5e-260)
       t_2
       (if (<= A 7e-126)
         (/ (* t_3 (- (sqrt (+ t_0 (+ A C))))) t_1)
         (if (<= A 2e-74)
           (pow (- (/ (sqrt (/ (* C -2.0) F)) (sqrt 2.0))) -1.0)
           (if (<= A 5.7e-12) t_5 (if (<= A 5.5e+107) t_2 t_5))))))))

C

double code(double A, double B, double C, double F) {
	return -sqrt(((2.0 * ((pow(B, 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B, 2.0)))))) / (pow(B, 2.0) - ((4.0 * A) * C));
}

↓

double code(double A, double B, double C, double F) {
	double t_0 = hypot(B, (A - C));
	double t_1 = fma(B, B, (A * (C * -4.0)));
	double t_2 = pow((t_1 / ((sqrt((2.0 * t_1)) * sqrt(F)) * -sqrt((C + (A + t_0))))), -1.0);
	double t_3 = sqrt((2.0 * (F * t_1)));
	double t_4 = sqrt(t_0);
	double t_5 = (t_3 * -sqrt(fma(t_4, t_4, (A + C)))) / t_1;
	double tmp;
	if (A <= -2.2e+24) {
		tmp = sqrt(2.0) * -sqrt((-0.5 * (F / A)));
	} else if (A <= 7.5e-260) {
		tmp = t_2;
	} else if (A <= 7e-126) {
		tmp = (t_3 * -sqrt((t_0 + (A + C)))) / t_1;
	} else if (A <= 2e-74) {
		tmp = pow(-(sqrt(((C * -2.0) / F)) / sqrt(2.0)), -1.0);
	} else if (A <= 5.7e-12) {
		tmp = t_5;
	} else if (A <= 5.5e+107) {
		tmp = t_2;
	} else {
		tmp = t_5;
	}
	return tmp;
}

Julia

function code(A, B, C, F)
	return Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C)) * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0))))))) / Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C)))
end

↓

function code(A, B, C, F)
	t_0 = hypot(B, Float64(A - C))
	t_1 = fma(B, B, Float64(A * Float64(C * -4.0)))
	t_2 = Float64(t_1 / Float64(Float64(sqrt(Float64(2.0 * t_1)) * sqrt(F)) * Float64(-sqrt(Float64(C + Float64(A + t_0)))))) ^ -1.0
	t_3 = sqrt(Float64(2.0 * Float64(F * t_1)))
	t_4 = sqrt(t_0)
	t_5 = Float64(Float64(t_3 * Float64(-sqrt(fma(t_4, t_4, Float64(A + C))))) / t_1)
	tmp = 0.0
	if (A <= -2.2e+24)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(-0.5 * Float64(F / A)))));
	elseif (A <= 7.5e-260)
		tmp = t_2;
	elseif (A <= 7e-126)
		tmp = Float64(Float64(t_3 * Float64(-sqrt(Float64(t_0 + Float64(A + C))))) / t_1);
	elseif (A <= 2e-74)
		tmp = Float64(-Float64(sqrt(Float64(Float64(C * -2.0) / F)) / sqrt(2.0))) ^ -1.0;
	elseif (A <= 5.7e-12)
		tmp = t_5;
	elseif (A <= 5.5e+107)
		tmp = t_2;
	else
		tmp = t_5;
	end
	return tmp
end

Wolfram

code[A_, B_, C_, F_] := N[((-N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[A_, B_, C_, F_] := Block[{t$95$0 = N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$1 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(t$95$1 / N[(N[(N[Sqrt[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[N[(C + N[(A + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(2.0 * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$3 * (-N[Sqrt[N[(t$95$4 * t$95$4 + N[(A + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[A, -2.2e+24], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(-0.5 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[A, 7.5e-260], t$95$2, If[LessEqual[A, 7e-126], N[(N[(t$95$3 * (-N[Sqrt[N[(t$95$0 + N[(A + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[A, 2e-74], N[Power[(-N[(N[Sqrt[N[(N[(C * -2.0), $MachinePrecision] / F), $MachinePrecision]], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), -1.0], $MachinePrecision], If[LessEqual[A, 5.7e-12], t$95$5, If[LessEqual[A, 5.5e+107], t$95$2, t$95$5]]]]]]]]]]]]

Error

Bits error versus A

Bits error versus B

Bits error versus C

Bits error versus F

Derivation

1. Split input into 5 regimes

2. if A < -2.20000000000000002e24

   1. Initial program 61.9

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 58.5

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

   3. Taylor expanded in C around inf 41.4

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\right)} \]

   4. Simplified 41.4

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}} \]

   if -2.20000000000000002e24 < A < 7.5000000000000005e-260 or 5.7000000000000003e-12 < A < 5.5000000000000003e107

   1. Initial program 47.0

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 44.4

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

   3. Applied egg-rr 40.1

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right)} \cdot \sqrt{\left(C + A\right) + \mathsf{hypot}\left(B, A - C\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   4. Applied egg-rr 38.3

      \[\leadsto \frac{-\color{blue}{\left({\left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}^{0.5} \cdot \sqrt{F}\right)} \cdot \sqrt{\left(C + A\right) + \mathsf{hypot}\left(B, A - C\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   5. Applied egg-rr 38.2

      \[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{F}\right) \cdot \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}\right)}^{-1}} \]

   if 7.5000000000000005e-260 < A < 7e-126

   1. Initial program 48.2

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 45.0

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

   3. Applied egg-rr 40.2

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right)} \cdot \sqrt{\left(C + A\right) + \mathsf{hypot}\left(B, A - C\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   if 7e-126 < A < 1.99999999999999992e-74

   1. Initial program 46.2

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 43.9

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

   3. Applied egg-rr 39.2

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right)} \cdot \sqrt{\left(C + A\right) + \mathsf{hypot}\left(B, A - C\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   4. Applied egg-rr 43.9

      \[\leadsto \frac{-\color{blue}{\left({\left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}^{0.5} \cdot \sqrt{F}\right)} \cdot \sqrt{\left(C + A\right) + \mathsf{hypot}\left(B, A - C\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   5. Applied egg-rr 43.9

      \[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{F}\right) \cdot \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}\right)}^{-1}} \]

   6. Taylor expanded in A around inf 50.1

      \[\leadsto {\color{blue}{\left(-1 \cdot \frac{\sqrt{-2 \cdot \frac{C}{F}}}{\sqrt{2}}\right)}}^{-1} \]

   7. Simplified 50.1

      \[\leadsto {\color{blue}{\left(-\frac{\sqrt{\frac{-2 \cdot C}{F}}}{\sqrt{2}}\right)}}^{-1} \]

   if 1.99999999999999992e-74 < A < 5.7000000000000003e-12 or 5.5000000000000003e107 < A

   1. Initial program 56.0

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 48.5

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

   3. Applied egg-rr 40.6

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right)} \cdot \sqrt{\left(C + A\right) + \mathsf{hypot}\left(B, A - C\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   4. Applied egg-rr 40.6

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right)} \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{hypot}\left(B, A - C\right)}, \sqrt{\mathsf{hypot}\left(B, A - C\right)}, A + C\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 40.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -2.2 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{-0.5 \cdot \frac{F}{A}}\right)\\ \mathbf{elif}\;A \leq 7.5 \cdot 10^{-260}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{F}\right) \cdot \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}\right)}^{-1}\\ \mathbf{elif}\;A \leq 7 \cdot 10^{-126}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \left(-\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;A \leq 2 \cdot 10^{-74}:\\ \;\;\;\;{\left(-\frac{\sqrt{\frac{C \cdot -2}{F}}}{\sqrt{2}}\right)}^{-1}\\ \mathbf{elif}\;A \leq 5.7 \cdot 10^{-12}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(\sqrt{\mathsf{hypot}\left(B, A - C\right)}, \sqrt{\mathsf{hypot}\left(B, A - C\right)}, A + C\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;A \leq 5.5 \cdot 10^{+107}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{F}\right) \cdot \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(\sqrt{\mathsf{hypot}\left(B, A - C\right)}, \sqrt{\mathsf{hypot}\left(B, A - C\right)}, A + C\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))