ABCF->ab-angle b

Average Error: 52.1 → 34.6

Time: 34.0s

Precision: binary64

Cost: 21128

\[ \begin{array}{c}[A, C] = \mathsf{sort}([A, C])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -3.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\\ \mathbf{elif}\;B \leq 54:\\ \;\;\;\;\sqrt{\left(2 \cdot t_0\right) \cdot \left(F \cdot \left(A + A\right)\right)} \cdot \frac{1}{-t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

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 (* C (* A -4.0)))))
   (if (<= B -3.5e-67)
     (* (/ (sqrt 2.0) B) (sqrt (* F (- A (hypot A B)))))
     (if (<= B 54.0)
       (* (sqrt (* (* 2.0 t_0) (* F (+ A A)))) (/ 1.0 (- t_0)))
       (* (sqrt (* F (- A (hypot B A)))) (/ (- (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, (C * (A * -4.0)));
	double tmp;
	if (B <= -3.5e-67) {
		tmp = (sqrt(2.0) / B) * sqrt((F * (A - hypot(A, B))));
	} else if (B <= 54.0) {
		tmp = sqrt(((2.0 * t_0) * (F * (A + A)))) * (1.0 / -t_0);
	} else {
		tmp = sqrt((F * (A - hypot(B, A)))) * (-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(C * Float64(A * -4.0)))
	tmp = 0.0
	if (B <= -3.5e-67)
		tmp = Float64(Float64(sqrt(2.0) / B) * sqrt(Float64(F * Float64(A - hypot(A, B)))));
	elseif (B <= 54.0)
		tmp = Float64(sqrt(Float64(Float64(2.0 * t_0) * Float64(F * Float64(A + A)))) * Float64(1.0 / Float64(-t_0)));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(B, A)))) * 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[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -3.5e-67], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 54.0], N[(N[Sqrt[N[(N[(2.0 * t$95$0), $MachinePrecision] * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / (-t$95$0)), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < -3.5e-67

   1. Initial program 52.5

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

   2. Taylor expanded in C around 0 63.4

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

   3. Simplified 62.8

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

   4. Applied egg-rr 47.9

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

   5. Taylor expanded in C around 0 50.4

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

   6. Simplified 34.9

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

   if -3.5e-67 < B < 54

   1. Initial program 50.2

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

   2. Applied egg-rr 43.5

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

   3. Taylor expanded in C around inf 35.2

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

   if 54 < B

   1. Initial program 55.0

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

   2. Applied egg-rr 52.9

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

   3. Applied egg-rr 48.5

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

   4. Taylor expanded in C around 0 51.1

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

   5. Simplified 48.4

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

   6. Taylor expanded in C around 0 51.6

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

   7. Simplified 33.1

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

4. Final simplification 34.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -3.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\\ \mathbf{elif}\;B \leq 54:\\ \;\;\;\;\sqrt{\left(2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \cdot \left(F \cdot \left(A + A\right)\right)} \cdot \frac{1}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternatives

Alternative 1
Error	34.6
Cost	21000

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -3.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\\ \mathbf{elif}\;B \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A + A\right) \cdot \left(F \cdot t_0\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 2
Error	38.2
Cost	20300

\[\begin{array}{l} \mathbf{if}\;B \leq -2.4 \cdot 10^{-107}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{-102}:\\ \;\;\;\;\frac{1}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)}\\ \mathbf{elif}\;B \leq 10^{+135}:\\ \;\;\;\;\frac{\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot F\right)} \cdot \left(-B\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 3
Error	37.9
Cost	20168

\[\begin{array}{l} \mathbf{if}\;B \leq -2.4 \cdot 10^{-107}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\\ \mathbf{elif}\;B \leq 8.2 \cdot 10^{-95}:\\ \;\;\;\;\frac{1}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 4
Error	41.7
Cost	19972

\[\begin{array}{l} \mathbf{if}\;B \leq -2.4 \cdot 10^{-107}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{-102}:\\ \;\;\;\;\frac{1}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot F\right)} \cdot \left(-B\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 5
Error	45.6
Cost	14472

\[\begin{array}{l} t_0 := \sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot F\right)}\\ \mathbf{if}\;B \leq -2.4 \cdot 10^{-107}:\\ \;\;\;\;\left(B \cdot t_0\right) \cdot \frac{1}{B \cdot B}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{-102}:\\ \;\;\;\;\frac{1}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 \cdot \left(-B\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 6
Error	47.4
Cost	14408

\[\begin{array}{l} t_0 := \sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot F\right)}\\ \mathbf{if}\;B \leq -2.4 \cdot 10^{-107}:\\ \;\;\;\;\left(B \cdot t_0\right) \cdot \frac{1}{B \cdot B}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{-102}:\\ \;\;\;\;\frac{1}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{\left(F \cdot C\right) \cdot \left(-16 \cdot \left(A \cdot A\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 \cdot \left(-B\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 7
Error	47.5
Cost	14344

\[\begin{array}{l} t_0 := \sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot F\right)}\\ \mathbf{if}\;B \leq -2.4 \cdot 10^{-107}:\\ \;\;\;\;\left(B \cdot t_0\right) \cdot \frac{1}{B \cdot B}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{-102}:\\ \;\;\;\;\frac{1}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{\left(F \cdot C\right) \cdot \left(-16 \cdot \left(A \cdot A\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 \cdot \left(-B\right)}{B \cdot B}\\ \end{array} \]

Alternative 8
Error	53.0
Cost	13956

\[\begin{array}{l} t_0 := \sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot F\right)}\\ \mathbf{if}\;B \leq -4.5 \cdot 10^{-146}:\\ \;\;\;\;\left(B \cdot t_0\right) \cdot \frac{1}{B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 \cdot \left(-B\right)}{B \cdot B}\\ \end{array} \]

Alternative 9
Error	58.0
Cost	13760

\[\frac{\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot F\right)} \cdot \left(-B\right)}{B \cdot B} \]

Alternative 10
Error	61.6
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022225 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :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))))