ABCF->ab-angle a

Average Error: 52.1 → 36.2

Time: 31.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \frac{-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}{t_0}\\ t_2 := {B}^{2} + C \cdot \left(A \cdot -4\right)\\ t_3 := -\frac{\sqrt{\left(2 \cdot \left(t_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t_2}\\ \mathbf{if}\;t_3 \leq -7.788787492606855 \cdot 10^{-220}:\\ \;\;\;\;\left(\sqrt{2 \cdot t_0} \cdot \sqrt{F}\right) \cdot t_1\\ \mathbf{elif}\;t_3 \leq 5.962983613386414 \cdot 10^{-193}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(F \cdot t_0\right)} \cdot \sqrt{2 \cdot C + \frac{{B}^{2}}{A} \cdot -0.5}}{t_0}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;t_1 \cdot {\left(e^{0.25 \cdot \left(\log \left(-2 \cdot \left({B}^{2} + -4 \cdot \left(A \cdot C\right)\right)\right) - \log \left(\frac{-1}{F}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

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 (/ (- (sqrt (+ C (+ A (hypot B (- A C)))))) t_0))
        (t_2 (+ (pow B 2.0) (* C (* A -4.0))))
        (t_3
         (-
          (/
           (sqrt
            (*
             (* 2.0 (* t_2 F))
             (+ (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
           t_2))))
   (if (<= t_3 -7.788787492606855e-220)
     (* (* (sqrt (* 2.0 t_0)) (sqrt F)) t_1)
     (if (<= t_3 5.962983613386414e-193)
       (-
        (/
         (*
          (sqrt (* 2.0 (* F t_0)))
          (sqrt (+ (* 2.0 C) (* (/ (pow B 2.0) A) -0.5))))
         t_0))
       (if (<= t_3 INFINITY)
         (*
          t_1
          (pow
           (exp
            (*
             0.25
             (-
              (log (* -2.0 (+ (pow B 2.0) (* -4.0 (* A C)))))
              (log (/ -1.0 F)))))
           2.0))
         (* (sqrt (* -0.5 (/ F C))) (- (sqrt 2.0))))))))

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 = -sqrt((C + (A + hypot(B, (A - C))))) / t_0;
	double t_2 = pow(B, 2.0) + (C * (A * -4.0));
	double t_3 = -(sqrt(((2.0 * (t_2 * F)) * ((A + C) + sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / t_2);
	double tmp;
	if (t_3 <= -7.788787492606855e-220) {
		tmp = (sqrt((2.0 * t_0)) * sqrt(F)) * t_1;
	} else if (t_3 <= 5.962983613386414e-193) {
		tmp = -((sqrt((2.0 * (F * t_0))) * sqrt(((2.0 * C) + ((pow(B, 2.0) / A) * -0.5)))) / t_0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_1 * pow(exp((0.25 * (log((-2.0 * (pow(B, 2.0) + (-4.0 * (A * C))))) - log((-1.0 / F))))), 2.0);
	} else {
		tmp = sqrt((-0.5 * (F / C))) * -sqrt(2.0);
	}
	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(Float64(-sqrt(Float64(C + Float64(A + hypot(B, Float64(A - C)))))) / t_0)
	t_2 = Float64((B ^ 2.0) + Float64(C * Float64(A * -4.0)))
	t_3 = Float64(-Float64(sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) + sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_2))
	tmp = 0.0
	if (t_3 <= -7.788787492606855e-220)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_0)) * sqrt(F)) * t_1);
	elseif (t_3 <= 5.962983613386414e-193)
		tmp = Float64(-Float64(Float64(sqrt(Float64(2.0 * Float64(F * t_0))) * sqrt(Float64(Float64(2.0 * C) + Float64(Float64((B ^ 2.0) / A) * -0.5)))) / t_0));
	elseif (t_3 <= Inf)
		tmp = Float64(t_1 * (exp(Float64(0.25 * Float64(log(Float64(-2.0 * Float64((B ^ 2.0) + Float64(-4.0 * Float64(A * C))))) - log(Float64(-1.0 / F))))) ^ 2.0));
	else
		tmp = Float64(sqrt(Float64(-0.5 * Float64(F / C))) * Float64(-sqrt(2.0)));
	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[((-N[Sqrt[N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B, 2.0], $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = (-N[(N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision])}, If[LessEqual[t$95$3, -7.788787492606855e-220], N[(N[(N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 5.962983613386414e-193], (-N[(N[(N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(2.0 * C), $MachinePrecision] + N[(N[(N[Power[B, 2.0], $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), If[LessEqual[t$95$3, Infinity], N[(t$95$1 * N[Power[N[Exp[N[(0.25 * N[(N[Log[N[(-2.0 * N[(N[Power[B, 2.0], $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(-0.5 * N[(F / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $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))) < -7.7887874926068554e-220

   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 31.2

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

   3. Applied egg-rr 23.0

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

   4. Applied egg-rr 22.5

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

   5. Applied egg-rr 12.5

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

   if -7.7887874926068554e-220 < (/.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))) < 5.96298361338641355e-193

   1. Initial program 61.5

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

   2. Simplified 59.4

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

   3. Applied egg-rr 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. Taylor expanded in A around -inf 47.8

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

   if 5.96298361338641355e-193 < (/.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 38.5

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

   2. Simplified 25.5

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

   3. Applied egg-rr 12.2

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

   4. Applied egg-rr 12.3

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

   5. Applied egg-rr 12.5

      \[\leadsto \frac{\color{blue}{{\left({\left(2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right)\right)}^{0.25}\right)}^{2}}}{1} \cdot \frac{-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Taylor expanded in F around -inf 8.2

      \[\leadsto \frac{{\color{blue}{\left(e^{0.25 \cdot \left(\log \left(-2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) - \log \left(\frac{-1}{F}\right)\right)}\right)}}^{2}}{1} \cdot \frac{-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   if +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 63.3

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

   3. Taylor expanded in A around inf 58.0

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

   4. Simplified 58.0

      \[\leadsto \color{blue}{-\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \sqrt{2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 36.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} + C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} + C \cdot \left(A \cdot -4\right)} \leq -7.788787492606855 \cdot 10^{-220}:\\ \;\;\;\;\left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;-\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} + C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} + C \cdot \left(A \cdot -4\right)} \leq 5.962983613386414 \cdot 10^{-193}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \sqrt{2 \cdot C + \frac{{B}^{2}}{A} \cdot -0.5}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;-\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} + C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} + C \cdot \left(A \cdot -4\right)} \leq \infty:\\ \;\;\;\;\frac{-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot {\left(e^{0.25 \cdot \left(\log \left(-2 \cdot \left({B}^{2} + -4 \cdot \left(A \cdot C\right)\right)\right) - \log \left(\frac{-1}{F}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(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))))