Toniolo and Linder, Equation (3b), real

Average Error: 3.9 → 3.0

Time: 20.7s

Precision: binary64

Cost: 84481

Math

\[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.9996203724830907:\\ \;\;\;\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin ky + 0.5 \cdot \frac{kx \cdot kx}{\sin ky}}\\ \end{array}\]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))

↓

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))
      0.9996203724830907)
   (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th))
   (* (sin th) (/ (sin ky) (+ (sin ky) (* 0.5 (/ (* kx kx) (sin ky))))))))

C

double code(double kx, double ky, double th) {
	return (sin(ky) / sqrt(pow(sin(kx), 2.0) + pow(sin(ky), 2.0))) * sin(th);
}

↓

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt(pow(sin(kx), 2.0) + pow(sin(ky), 2.0))) <= 0.9996203724830907) {
		tmp = (sin(ky) / sqrt(pow(sin(kx), 2.0) + pow(sin(ky), 2.0))) * sin(th);
	} else {
		tmp = sin(th) * (sin(ky) / (sin(ky) + (0.5 * ((kx * kx) / sin(ky)))));
	}
	return tmp;
}

Error

Bits error versus kx

Bits error versus ky

Bits error versus th

Alternatives

Alternative 1
Error	4.5
Cost	161600

\[\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt[3]{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sqrt[3]{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \left(\sin th \cdot \frac{\sqrt[3]{\sin ky}}{\sqrt[3]{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}\right)\]

Alternative 2
Error	4.9
Cost	155200

\[\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt{\sqrt[3]{{\sin kx}^{2} + {\sin ky}^{2}} \cdot \sqrt[3]{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \left(\sin th \cdot \frac{\sqrt[3]{\sin ky}}{\sqrt{\sqrt[3]{{\sin kx}^{2} + {\sin ky}^{2}}}}\right)\]

Alternative 3
Error	4.4
Cost	142400

\[\left(\sqrt[3]{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}\right) \cdot \left(\sin th \cdot \sqrt[3]{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}\right)\]

Alternative 4
Error	34.2
Cost	142272

\[\frac{\sqrt{\sin ky}}{\sqrt{\sqrt[3]{{\sin kx}^{2} + {\sin ky}^{2}} \cdot \sqrt[3]{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \left(\sin th \cdot \frac{\sqrt{\sin ky}}{\sqrt{\sqrt[3]{{\sin kx}^{2} + {\sin ky}^{2}}}}\right)\]

Alternative 5
Error	5.6
Cost	129472

\[\frac{1}{\sqrt[3]{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sqrt[3]{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \left(\sin th \cdot \frac{\sin ky}{\sqrt[3]{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}\right)\]

Alternative 6
Error	4.8
Cost	129344

\[\sin th \cdot \frac{\sin ky}{\sqrt[3]{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \left(\sqrt[3]{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sqrt[3]{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\]

Alternative 7
Error	49.0
Cost	129216

\[\left(\sqrt{\sin th} \cdot \frac{\sqrt{\sin ky}}{\sqrt{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}\right) \cdot \left(\sqrt{\sin th} \cdot \frac{\sqrt{\sin ky}}{\sqrt{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}\right)\]

Alternative 8
Error	5.2
Cost	123072

\[\frac{1}{\sqrt{\sqrt[3]{{\sin kx}^{2} + {\sin ky}^{2}} \cdot \sqrt[3]{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \left(\sin th \cdot \frac{\sin ky}{\sqrt{\sqrt[3]{{\sin kx}^{2} + {\sin ky}^{2}}}}\right)\]

Alternative 9
Error	4.9
Cost	122816

\[\frac{\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}}{\sqrt{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \left(\sin th \cdot \frac{\sqrt[3]{\sin ky}}{\sqrt{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}\right)\]

Alternative 10
Error	11.4
Cost	116672

\[\frac{\sin ky}{\sqrt{{\left({\sin kx}^{2}\right)}^{3} + {\left({\sin ky}^{2}\right)}^{3}}} \cdot \left(\sin th \cdot \sqrt{{\sin kx}^{4} + \left({\sin ky}^{4} - {\sin kx}^{2} \cdot {\sin ky}^{2}\right)}\right)\]

Alternative 11
Error	33.9
Cost	109888

\[\frac{\sqrt{\sin ky}}{\sqrt{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \left(\sin th \cdot \frac{\sqrt{\sin ky}}{\sqrt{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}\right)\]

Alternative 12
Error	11.1
Cost	103744

\[\sin th \cdot \frac{\sin ky}{\frac{\sqrt{{\sin kx}^{6} + {\sin ky}^{6}}}{\sqrt{{\sin kx}^{4} + \left({\sin ky}^{4} - {\sin kx}^{2} \cdot {\sin ky}^{2}\right)}}}\]

Alternative 13
Error	36.7
Cost	103744

\[\frac{\sin ky}{\sqrt{{\sin kx}^{2} \cdot {\sin kx}^{2} - {\sin ky}^{2} \cdot {\sin ky}^{2}}} \cdot \left(\sin th \cdot \sqrt{{\sin kx}^{2} - {\sin ky}^{2}}\right)\]

Alternative 14
Error	32.0
Cost	103616

\[\sqrt{\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sqrt{\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}\]

Alternative 15
Error	33.9
Cost	97088

\[\sqrt{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \left(\sin th \cdot \sqrt{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}\right)\]

Alternative 16
Error	4.2
Cost	90688

\[\sin ky \cdot \left(\frac{\sin th}{\sqrt{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \frac{1}{\sqrt{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}\right)\]

Alternative 17
Error	4.8
Cost	90688

\[\frac{1}{\sqrt{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \left(\sin th \cdot \frac{\sin ky}{\sqrt{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}\right)\]

Alternative 18
Error	4.5
Cost	90560

\[\sin th \cdot \frac{\sin ky}{\left|\sqrt[3]{{\sin kx}^{2} + {\sin ky}^{2}}\right| \cdot \sqrt{\sqrt[3]{{\sin kx}^{2} + {\sin ky}^{2}}}}\]

Alternative 19
Error	4.2
Cost	90560

\[\sin th \cdot \frac{\sin ky}{\sqrt{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sqrt{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}\]

Alternative 20
Error	4.2
Cost	90560

\[\frac{\sin ky}{\sqrt{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \frac{\sin th}{\sqrt{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}\]

Alternative 21
Error	4.9
Cost	77632

\[\sqrt[3]{\sin th} \cdot \left(\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \left(\sqrt[3]{\sin th} \cdot \sqrt[3]{\sin th}\right)\right)\]

Alternative 22
Error	4.8
Cost	77632

\[\left(\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}\right) \cdot \left(\sin th \cdot \frac{\sqrt[3]{\sin ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\]

Alternative 23
Error	34.4
Cost	71168

\[\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\left({\sin ky}^{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right)}\right)}^{\left(\sqrt[3]{2}\right)}}}\]

Alternative 24
Error	4.4
Cost	71168

\[\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\left(\sqrt[3]{\sin ky}\right)}^{4} \cdot {\left(\sqrt[3]{\sin ky}\right)}^{2}}}\]

Alternative 25
Error	4.8
Cost	71168

\[{\left(\sqrt[3]{\sin ky}\right)}^{2} \cdot \frac{\sqrt[3]{\sin ky}}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin th}}\]

Alternative 26
Error	33.6
Cost	64704

\[\sqrt{\sin th} \cdot \frac{\sin ky}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sqrt{\sin th}}}\]

Alternative 27
Error	33.6
Cost	64704

\[\sqrt{\sin th} \cdot \left(\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sqrt{\sin th}\right)\]

Alternative 28
Error	34.0
Cost	64704

\[\sqrt{\sin ky} \cdot \left(\sin th \cdot \frac{\sqrt{\sin ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\]

Alternative 29
Error	34.2
Cost	64640

\[\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\left({\sin ky}^{\left(\sqrt{2}\right)}\right)}^{\left(\sqrt{2}\right)}}}\]

Alternative 30
Error	40.6
Cost	64512

\[\log \left({\left(e^{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}\right)}^{\sin ky}\right)\]

Alternative 31
Error	7.3
Cost	58240

\[\sin th \cdot \frac{\sin ky}{\sqrt[3]{{\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}^{3}}}\]

Alternative 32
Error	18.6
Cost	58240

\[\sin ky \cdot \sqrt[3]{{\left(\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}^{3}}\]

Alternative 33
Error	27.4
Cost	58240

\[\sqrt[3]{{\left(\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}^{3}}\]

Alternative 34
Error	9.9
Cost	58176

\[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + \log \left(e^{{\sin ky}^{2}}\right)}} \cdot \sin th\]

Alternative 35
Error	15.5
Cost	58176

\[\sin th \cdot \frac{\sin ky}{\log \left(e^{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\]

Alternative 36
Error	15.4
Cost	58176

\[\sin ky \cdot \frac{\sin th}{\log \left(e^{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}\]

Alternative 37
Error	37.0
Cost	58176

\[\log \left(e^{\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}\right)\]

Alternative 38
Error	9.9
Cost	58176

\[\sin th \cdot \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \log \left(e^{{\sin kx}^{2}}\right)}}\]

Alternative 39
Error	33.9
Cost	51840

\[\sin th \cdot \frac{\sin ky}{\sqrt{{\sin ky}^{2} + e^{2 \cdot \log \sin kx}}}\]

Alternative 40
Error	7.4
Cost	51776

\[\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + \sqrt[3]{{\sin ky}^{6}}}}\]

Alternative 41
Error	7.4
Cost	51776

\[\sin th \cdot \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \sqrt[3]{{\sin kx}^{6}}}}\]

Alternative 42
Error	7.4
Cost	51776

\[\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \sqrt[3]{{\sin ky}^{6}}}}\]

Alternative 43
Error	4.1
Cost	45632

\[\sin ky \cdot \frac{\frac{1}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}{\frac{1}{\sin th}}\]

Alternative 44
Error	4.0
Cost	45504

\[\sin ky \cdot \left(\sin th \cdot \frac{1}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\]

Alternative 45
Error	4.0
Cost	45504

\[\sin ky \cdot \frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin th}}\]

Alternative 46
Error	4.3
Cost	45504

\[\sin ky \cdot \left(\sin th \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\]

Alternative 47
Error	5.4
Cost	45504

\[\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \left(\sin ky \cdot \sin th\right)\]

Alternative 48
Error	4.0
Cost	45504

\[\frac{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}{\frac{1}{\sin th}}\]

Alternative 49
Error	3.9
Cost	45376

\[\frac{\sin ky}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin th}}\]

Alternative 50
Error	3.9
Cost	45376

\[\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\]

Alternative 51
Error	3.9
Cost	45376

\[\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\]

Alternative 52
Error	5.1
Cost	45376

\[\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\]

Alternative 53
Error	33.5
Cost	39104

\[\sin ky \cdot \left(th \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\]

Alternative 54
Error	34.6
Cost	39104

\[\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \left(\sin ky \cdot th\right)\]

Alternative 55
Error	34.1
Cost	32768

\[\sin ky \cdot \frac{1}{\frac{\sqrt{{\sin kx}^{2} + ky \cdot ky}}{\sin th}}\]

Alternative 56
Error	34.1
Cost	32640

\[\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + ky \cdot ky}}\]

Alternative 57
Error	30.2
Cost	32640

\[\sin th \cdot \frac{\sin ky}{\sqrt{{\sin ky}^{2} + kx \cdot kx}}\]

Alternative 58
Error	34.1
Cost	32640

\[\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + ky \cdot ky}}\]

Alternative 59
Error	34.1
Cost	32640

\[\frac{\sin ky}{\frac{\sqrt{{\sin kx}^{2} + ky \cdot ky}}{\sin th}}\]

Alternative 60
Error	47.4
Cost	26432

\[\sin th \cdot \frac{\sin ky}{\sin kx + 0.5 \cdot \frac{ky \cdot ky}{\sin kx}}\]

Alternative 61
Error	45.1
Cost	26432

\[\sin th \cdot \frac{\sin ky}{\sin ky + 0.5 \cdot \frac{kx \cdot kx}{\sin ky}}\]

Alternative 62
Error	46.5
Cost	19648

\[\sin ky \cdot \frac{1}{\frac{\sin kx}{\sin th}}\]

Alternative 63
Error	48.9
Cost	19648

\[\sin ky \cdot \frac{1}{\frac{\sin ky}{\sin th}}\]

Alternative 64
Error	46.5
Cost	19520

\[\sin ky \cdot \frac{\sin th}{\sin kx}\]

Alternative 65
Error	46.5
Cost	19520

\[\sin th \cdot \frac{\sin ky}{\sin kx}\]

Alternative 66
Error	48.9
Cost	19520

\[\sin th \cdot \frac{\sin ky}{\sin ky}\]

Alternative 67
Error	46.5
Cost	19520

\[\frac{\sin ky}{\frac{\sin kx}{\sin th}}\]

Alternative 68
Error	48.1
Cost	13120

\[ky \cdot \frac{\sin th}{\sin kx}\]

Alternative 69
Error	48.7
Cost	13120

\[\frac{ky \cdot \sin th}{\sin kx}\]

Alternative 70
Error	48.9
Cost	6464

\[\sin th\]

Alternative 71
Error	60.3
Cost	64

\[1\]

Alternative 72
Error	56.2
Cost	64

\[0\]

Alternative 73
Error	60.2
Cost	64

\[-1\]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) 2) (pow.f64 (sin.f64 ky) 2)))) < 0.999620372483090658

   1. Initial program 2.5

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\]

   2. Simplified 2.5

      \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}\]

   if 0.999620372483090658 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) 2) (pow.f64 (sin.f64 ky) 2))))

   1. Initial program 9.2

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\]

   2. Taylor expanded around 0 4.8

      \[\leadsto \frac{\sin ky}{\color{blue}{0.5 \cdot \frac{{kx}^{2}}{\sin ky} + \sin ky}} \cdot \sin th\]

   3. Simplified 4.8

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin ky + 0.5 \cdot \frac{kx \cdot kx}{\sin ky}}} \cdot \sin th\]

   4. Simplified 4.8

      \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sin ky + 0.5 \cdot \frac{kx \cdot kx}{\sin ky}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 3.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.9996203724830907:\\ \;\;\;\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin ky + 0.5 \cdot \frac{kx \cdot kx}{\sin ky}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021042 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  :precision binary64
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))