ABCF->ab-angle a

Average Error: 52.1 → 35.5

Time: 29.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := 2 \cdot t_0\\ t_2 := \mathsf{hypot}\left(B, A - C\right)\\ t_3 := {B}^{2} + C \cdot \left(A \cdot -4\right)\\ t_4 := \frac{-\sqrt{\left(2 \cdot \left(t_3 \cdot F\right)\right) \cdot \left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} + \left(A + C\right)\right)}}{t_3}\\ \mathbf{if}\;t_4 \leq -4.325573197083108 \cdot 10^{-208}:\\ \;\;\;\;\left(\left(\sqrt{t_1} \cdot \sqrt{F}\right) \cdot \sqrt{C + \left(A + t_2\right)}\right) \cdot \frac{-1}{t_0}\\ \mathbf{elif}\;t_4 \leq 0:\\ \;\;\;\;\frac{\sqrt{2 \cdot A + \frac{{B}^{2}}{C} \cdot -0.5} \cdot \left(\sqrt{F} \cdot \left(-{t_1}^{0.5}\right)\right)}{t_0}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{t_2 + \left(A + C\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

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 (fma B B (* A (* C -4.0))))
        (t_1 (* 2.0 t_0))
        (t_2 (hypot B (- A C)))
        (t_3 (+ (pow B 2.0) (* C (* A -4.0))))
        (t_4
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_3 F))
             (+ (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))) (+ A C)))))
          t_3)))
   (if (<= t_4 -4.325573197083108e-208)
     (* (* (* (sqrt t_1) (sqrt F)) (sqrt (+ C (+ A t_2)))) (/ -1.0 t_0))
     (if (<= t_4 0.0)
       (/
        (*
         (sqrt (+ (* 2.0 A) (* (/ (pow B 2.0) C) -0.5)))
         (* (sqrt F) (- (pow t_1 0.5))))
        t_0)
       (if (<= t_4 INFINITY)
         (-
          (/
           (* (sqrt (* 2.0 (* F (* -4.0 (* A C))))) (sqrt (+ t_2 (+ A C))))
           t_0))
         (* (sqrt (* F (+ C (hypot C B)))) (/ (- (sqrt 2.0)) 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 = fma(B, B, (A * (C * -4.0)));
	double t_1 = 2.0 * t_0;
	double t_2 = hypot(B, (A - C));
	double t_3 = pow(B, 2.0) + (C * (A * -4.0));
	double t_4 = -sqrt(((2.0 * (t_3 * F)) * (sqrt((pow(B, 2.0) + pow((A - C), 2.0))) + (A + C)))) / t_3;
	double tmp;
	if (t_4 <= -4.325573197083108e-208) {
		tmp = ((sqrt(t_1) * sqrt(F)) * sqrt((C + (A + t_2)))) * (-1.0 / t_0);
	} else if (t_4 <= 0.0) {
		tmp = (sqrt(((2.0 * A) + ((pow(B, 2.0) / C) * -0.5))) * (sqrt(F) * -pow(t_1, 0.5))) / t_0;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = -((sqrt((2.0 * (F * (-4.0 * (A * C))))) * sqrt((t_2 + (A + C)))) / t_0);
	} else {
		tmp = sqrt((F * (C + hypot(C, B)))) * (-sqrt(2.0) / 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 = fma(B, B, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(2.0 * t_0)
	t_2 = hypot(B, Float64(A - C))
	t_3 = Float64((B ^ 2.0) + Float64(C * Float64(A * -4.0)))
	t_4 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_3 * F)) * Float64(sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0))) + Float64(A + C))))) / t_3)
	tmp = 0.0
	if (t_4 <= -4.325573197083108e-208)
		tmp = Float64(Float64(Float64(sqrt(t_1) * sqrt(F)) * sqrt(Float64(C + Float64(A + t_2)))) * Float64(-1.0 / t_0));
	elseif (t_4 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * A) + Float64(Float64((B ^ 2.0) / C) * -0.5))) * Float64(sqrt(F) * Float64(-(t_1 ^ 0.5)))) / t_0);
	elseif (t_4 <= Inf)
		tmp = Float64(-Float64(Float64(sqrt(Float64(2.0 * Float64(F * Float64(-4.0 * Float64(A * C))))) * sqrt(Float64(t_2 + Float64(A + C)))) / t_0));
	else
		tmp = Float64(sqrt(Float64(F * Float64(C + hypot(C, B)))) * Float64(Float64(-sqrt(2.0)) / 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[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[B, 2.0], $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$3 * F), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(A + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, -4.325573197083108e-208], N[(N[(N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(C + N[(A + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(N[(N[Sqrt[N[(N[(2.0 * A), $MachinePrecision] + N[(N[(N[Power[B, 2.0], $MachinePrecision] / C), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Power[t$95$1, 0.5], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[t$95$4, Infinity], (-N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$2 + N[(A + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), N[(N[Sqrt[N[(F * N[(C + N[Sqrt[C ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / 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 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -4.32557319708310804e-208

   1. Initial program 37.5

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

   2. Simplified 32.1

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

   3. Applied egg-rr 23.8

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

   4. Applied egg-rr 16.6

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

   5. Applied egg-rr 15.9

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

   if -4.32557319708310804e-208 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -0.0

   1. Initial program 61.2

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

   2. Simplified 59.1

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

   3. Applied egg-rr 60.0

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

   4. Applied egg-rr 53.7

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

   5. Taylor expanded in C around -inf 47.6

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

   if -0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < +inf.0

   1. Initial program 39.0

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

   2. Simplified 26.5

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

   3. Applied egg-rr 11.4

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

   4. Taylor expanded in B around 0 11.9

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

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))

   1. Initial program 64.0

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

   2. Simplified 63.4

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

   3. Applied egg-rr 64.0

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

   4. Taylor expanded in A around 0 63.6

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

   5. Simplified 52.8

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

4. Final simplification 35.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} + C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right) \cdot \left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} + \left(A + C\right)\right)}}{{B}^{2} + C \cdot \left(A \cdot -4\right)} \leq -4.325573197083108 \cdot 10^{-208}:\\ \;\;\;\;\left(\left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{F}\right) \cdot \sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right) \cdot \frac{-1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} + C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right) \cdot \left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} + \left(A + C\right)\right)}}{{B}^{2} + C \cdot \left(A \cdot -4\right)} \leq 0:\\ \;\;\;\;\frac{\sqrt{2 \cdot A + \frac{{B}^{2}}{C} \cdot -0.5} \cdot \left(\sqrt{F} \cdot \left(-{\left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}^{0.5}\right)\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} + C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right) \cdot \left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} + \left(A + C\right)\right)}}{{B}^{2} + C \cdot \left(A \cdot -4\right)} \leq \infty:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

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))))