ABCF->ab-angle a

Average Error: 52.3 → 39.8

Time: 31.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := A + \mathsf{hypot}\left(B, A - C\right)\\ t_1 := C + t_0\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_3 := 2 \cdot \left(F \cdot t_2\right)\\ t_4 := \left(\sqrt{2 \cdot t_2} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{t_1}}{t_2}\\ t_5 := -\sqrt{2}\\ t_6 := \sqrt{-0.5 \cdot \frac{F}{C}} \cdot t_5\\ \mathbf{if}\;B \leq -8.738524181400431 \cdot 10^{+158}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;B \leq -435814139245693630:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 7.082312261175921 \cdot 10^{-228}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(1, C, t_0\right)} \cdot \left(-\sqrt{t_3}\right)}{t_2}\\ \mathbf{elif}\;B \leq 7.395390810525929 \cdot 10^{-138}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;B \leq 742.5427233374058:\\ \;\;\;\;-\frac{\sqrt{t_1 \cdot t_3}}{t_2}\\ \mathbf{elif}\;B \leq 8.49625271680864 \cdot 10^{+138}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \frac{t_5}{B}\\ \end{array} \]

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 (+ A (hypot B (- A C))))
        (t_1 (+ C t_0))
        (t_2 (fma B B (* A (* C -4.0))))
        (t_3 (* 2.0 (* F t_2)))
        (t_4 (* (* (sqrt (* 2.0 t_2)) (sqrt F)) (/ (- (sqrt t_1)) t_2)))
        (t_5 (- (sqrt 2.0)))
        (t_6 (* (sqrt (* -0.5 (/ F C))) t_5)))
   (if (<= B -8.738524181400431e+158)
     t_6
     (if (<= B -435814139245693630.0)
       t_4
       (if (<= B 7.082312261175921e-228)
         (/ (* (sqrt (fma 1.0 C t_0)) (- (sqrt t_3))) t_2)
         (if (<= B 7.395390810525929e-138)
           t_6
           (if (<= B 742.5427233374058)
             (- (/ (sqrt (* t_1 t_3)) t_2))
             (if (<= B 8.49625271680864e+138)
               t_4
               (* (sqrt (* F (+ C (hypot C B)))) (/ t_5 B))))))))))

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 = A + hypot(B, (A - C));
	double t_1 = C + t_0;
	double t_2 = fma(B, B, (A * (C * -4.0)));
	double t_3 = 2.0 * (F * t_2);
	double t_4 = (sqrt((2.0 * t_2)) * sqrt(F)) * (-sqrt(t_1) / t_2);
	double t_5 = -sqrt(2.0);
	double t_6 = sqrt((-0.5 * (F / C))) * t_5;
	double tmp;
	if (B <= -8.738524181400431e+158) {
		tmp = t_6;
	} else if (B <= -435814139245693630.0) {
		tmp = t_4;
	} else if (B <= 7.082312261175921e-228) {
		tmp = (sqrt(fma(1.0, C, t_0)) * -sqrt(t_3)) / t_2;
	} else if (B <= 7.395390810525929e-138) {
		tmp = t_6;
	} else if (B <= 742.5427233374058) {
		tmp = -(sqrt((t_1 * t_3)) / t_2);
	} else if (B <= 8.49625271680864e+138) {
		tmp = t_4;
	} else {
		tmp = sqrt((F * (C + hypot(C, B)))) * (t_5 / B);
	}
	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 = Float64(A + hypot(B, Float64(A - C)))
	t_1 = Float64(C + t_0)
	t_2 = fma(B, B, Float64(A * Float64(C * -4.0)))
	t_3 = Float64(2.0 * Float64(F * t_2))
	t_4 = Float64(Float64(sqrt(Float64(2.0 * t_2)) * sqrt(F)) * Float64(Float64(-sqrt(t_1)) / t_2))
	t_5 = Float64(-sqrt(2.0))
	t_6 = Float64(sqrt(Float64(-0.5 * Float64(F / C))) * t_5)
	tmp = 0.0
	if (B <= -8.738524181400431e+158)
		tmp = t_6;
	elseif (B <= -435814139245693630.0)
		tmp = t_4;
	elseif (B <= 7.082312261175921e-228)
		tmp = Float64(Float64(sqrt(fma(1.0, C, t_0)) * Float64(-sqrt(t_3))) / t_2);
	elseif (B <= 7.395390810525929e-138)
		tmp = t_6;
	elseif (B <= 742.5427233374058)
		tmp = Float64(-Float64(sqrt(Float64(t_1 * t_3)) / t_2));
	elseif (B <= 8.49625271680864e+138)
		tmp = t_4;
	else
		tmp = Float64(sqrt(Float64(F * Float64(C + hypot(C, B)))) * Float64(t_5 / B));
	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[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(C + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[(F * t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Sqrt[N[(2.0 * t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[((-N[Sqrt[t$95$1], $MachinePrecision]) / t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$6 = N[(N[Sqrt[N[(-0.5 * N[(F / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$5), $MachinePrecision]}, If[LessEqual[B, -8.738524181400431e+158], t$95$6, If[LessEqual[B, -435814139245693630.0], t$95$4, If[LessEqual[B, 7.082312261175921e-228], N[(N[(N[Sqrt[N[(1.0 * C + t$95$0), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[t$95$3], $MachinePrecision])), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[B, 7.395390810525929e-138], t$95$6, If[LessEqual[B, 742.5427233374058], (-N[(N[Sqrt[N[(t$95$1 * t$95$3), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]), If[LessEqual[B, 8.49625271680864e+138], t$95$4, N[(N[Sqrt[N[(F * N[(C + N[Sqrt[C ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$5 / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]

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 B < -8.73852418140043132e158 or 7.082312261175921e-228 < B < 7.39539081052592889e-138

   1. Initial program 59.4

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

   2. Simplified 56.7

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

   3. Taylor expanded in A around inf 57.1

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

   4. Simplified 57.1

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

   if -8.73852418140043132e158 < B < -435814139245693632 or 742.542723337405846 < B < 8.4962527168086406e138

   1. Initial program 45.9

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

   2. Simplified 42.2

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

   3. Applied egg-rr 34.9

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

   4. Applied egg-rr 34.8

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

   5. Applied egg-rr 25.9

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

   if -435814139245693632 < B < 7.082312261175921e-228

   1. Initial program 49.6

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

   2. Simplified 44.3

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

   3. Applied egg-rr 40.9

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

   4. Applied egg-rr 40.2

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

   if 7.39539081052592889e-138 < B < 742.542723337405846

   1. Initial program 46.5

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

   2. Simplified 41.5

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

   3. Applied egg-rr 36.5

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

   4. Applied egg-rr 41.5

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

   if 8.4962527168086406e138 < B

   1. Initial program 63.1

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

   2. Simplified 62.9

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

   3. Applied egg-rr 61.6

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

   4. Applied egg-rr 61.6

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

   5. Applied egg-rr 61.6

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

   6. Taylor expanded in A around 0 61.3

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

   7. Simplified 34.4

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

4. Final simplification 39.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -8.738524181400431 \cdot 10^{+158}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq -435814139245693630:\\ \;\;\;\;\left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 7.082312261175921 \cdot 10^{-228}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(1, C, A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 7.395390810525929 \cdot 10^{-138}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 742.5427233374058:\\ \;\;\;\;-\frac{\sqrt{\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 8.49625271680864 \cdot 10^{+138}:\\ \;\;\;\;\left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(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))))