tanhf (example 3.4)

Average Error: 30.4 → 0.8

Time: 18.7s

Precision: binary64

Cost: 45764

Math

\[\frac{1 - \cos x}{\sin x} \]

↓

\[\begin{array}{l} t_0 := \frac{1 - \cos x}{\sin x}\\ \mathbf{if}\;t_0 \leq -0.02:\\ \;\;\;\;\frac{\cos x - \cos x \cdot \cos x}{\cos x \cdot \sin x}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;0.5 \cdot x + 0.041666666666666664 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (sin x)))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 (cos x)) (sin x))))
   (if (<= t_0 -0.02)
     (/ (- (cos x) (* (cos x) (cos x))) (* (cos x) (sin x)))
     (if (<= t_0 2e-7)
       (+ (* 0.5 x) (* 0.041666666666666664 (pow x 3.0)))
       t_0))))

C

double code(double x) {
	return (1.0 - cos(x)) / sin(x);
}

↓

double code(double x) {
	double t_0 = (1.0 - cos(x)) / sin(x);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = (cos(x) - (cos(x) * cos(x))) / (cos(x) * sin(x));
	} else if (t_0 <= 2e-7) {
		tmp = (0.5 * x) + (0.041666666666666664 * pow(x, 3.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - cos(x)) / sin(x)
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - cos(x)) / sin(x)
    if (t_0 <= (-0.02d0)) then
        tmp = (cos(x) - (cos(x) * cos(x))) / (cos(x) * sin(x))
    else if (t_0 <= 2d-7) then
        tmp = (0.5d0 * x) + (0.041666666666666664d0 * (x ** 3.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	return (1.0 - Math.cos(x)) / Math.sin(x);
}

↓

public static double code(double x) {
	double t_0 = (1.0 - Math.cos(x)) / Math.sin(x);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = (Math.cos(x) - (Math.cos(x) * Math.cos(x))) / (Math.cos(x) * Math.sin(x));
	} else if (t_0 <= 2e-7) {
		tmp = (0.5 * x) + (0.041666666666666664 * Math.pow(x, 3.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	return (1.0 - math.cos(x)) / math.sin(x)

↓

def code(x):
	t_0 = (1.0 - math.cos(x)) / math.sin(x)
	tmp = 0
	if t_0 <= -0.02:
		tmp = (math.cos(x) - (math.cos(x) * math.cos(x))) / (math.cos(x) * math.sin(x))
	elif t_0 <= 2e-7:
		tmp = (0.5 * x) + (0.041666666666666664 * math.pow(x, 3.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / sin(x))
end

↓

function code(x)
	t_0 = Float64(Float64(1.0 - cos(x)) / sin(x))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(Float64(cos(x) - Float64(cos(x) * cos(x))) / Float64(cos(x) * sin(x)));
	elseif (t_0 <= 2e-7)
		tmp = Float64(Float64(0.5 * x) + Float64(0.041666666666666664 * (x ^ 3.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - cos(x)) / sin(x);
end

↓

function tmp_2 = code(x)
	t_0 = (1.0 - cos(x)) / sin(x);
	tmp = 0.0;
	if (t_0 <= -0.02)
		tmp = (cos(x) - (cos(x) * cos(x))) / (cos(x) * sin(x));
	elseif (t_0 <= 2e-7)
		tmp = (0.5 * x) + (0.041666666666666664 * (x ^ 3.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]

↓

code[x_] := Block[{t$95$0 = N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], N[(N[(N[Cos[x], $MachinePrecision] - N[(N[Cos[x], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-7], N[(N[(0.5 * x), $MachinePrecision] + N[(0.041666666666666664 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Target

Original	30.4
Target	0.0
Herbie	0.8

\[\tan \left(\frac{x}{2}\right) \]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 1 (cos.f64 x)) (sin.f64 x)) < -0.0200000000000000004

   1. Initial program 0.8

      \[\frac{1 - \cos x}{\sin x} \]

   2. Applied egg-rr 0.8

      \[\leadsto \frac{\color{blue}{\frac{\cos x \cdot \left(1 - \cos x\right)}{\cos x}}}{\sin x} \]

   3. Applied egg-rr 0.9

      \[\leadsto \frac{\frac{\color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\cos x \ne 0:\\ \;\;\;\;\frac{\left(1 - \cos x\right) \cdot \left(\cos x \cdot \cos x\right)}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(1 - \cos x\right)\\ } \end{array}}}{\cos x}}{\sin x} \]

   4. Taylor expanded in x around inf 0.8

      \[\leadsto \color{blue}{\frac{\begin{array}{l} \mathbf{if}\;\cos x \ne 0:\\ \;\;\;\;\cos x \cdot \left(1 - \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(1 - \cos x\right)\\ \end{array}}{\cos x \cdot \sin x}} \]

   5. Simplified 0.8

      \[\leadsto \color{blue}{\frac{\cos x \cdot \left(1 - \cos x\right)}{\cos x \cdot \sin x}} \]
      Proof

      [Start] 0.8	\[ \frac{\begin{array}{l} \mathbf{if}\;\cos x \ne 0:\\ \;\;\;\;\cos x \cdot \left(1 - \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(1 - \cos x\right)\\ \end{array}}{\cos x \cdot \sin x} \]
      rational_best-simplify-5 [=>] 0.8	\[ \frac{\color{blue}{\cos x \cdot \left(1 - \cos x\right)}}{\cos x \cdot \sin x} \]

   6. Applied egg-rr 0.9

      \[\leadsto \frac{\color{blue}{\cos x + \cos x \cdot \left(-\cos x\right)}}{\cos x \cdot \sin x} \]

   7. Simplified 0.9

      \[\leadsto \frac{\color{blue}{\cos x - \cos x \cdot \cos x}}{\cos x \cdot \sin x} \]
      Proof

      [Start] 0.9	\[ \frac{\cos x + \cos x \cdot \left(-\cos x\right)}{\cos x \cdot \sin x} \]
      rational_best-simplify-53 [=>] 0.9	\[ \frac{\cos x + \color{blue}{\left(-\cos x \cdot \cos x\right)}}{\cos x \cdot \sin x} \]
      rational_best-simplify-61 [<=] 0.9	\[ \frac{\color{blue}{\cos x - \cos x \cdot \cos x}}{\cos x \cdot \sin x} \]

   if -0.0200000000000000004 < (/.f64 (-.f64 1 (cos.f64 x)) (sin.f64 x)) < 1.9999999999999999e-7

   1. Initial program 59.9

      \[\frac{1 - \cos x}{\sin x} \]

   2. Taylor expanded in x around 0 0.4

      \[\leadsto \color{blue}{0.5 \cdot x + 0.041666666666666664 \cdot {x}^{3}} \]

   if 1.9999999999999999e-7 < (/.f64 (-.f64 1 (cos.f64 x)) (sin.f64 x))

   1. Initial program 1.4

      \[\frac{1 - \cos x}{\sin x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \leq -0.02:\\ \;\;\;\;\frac{\cos x - \cos x \cdot \cos x}{\cos x \cdot \sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;0.5 \cdot x + 0.041666666666666664 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{\sin x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.7
Cost	39496

\[\begin{array}{l} t_0 := \frac{1 - \cos x}{\sin x}\\ \mathbf{if}\;t_0 \leq -0.02:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;0.5 \cdot x + 0.041666666666666664 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.8
Cost	39496

\[\begin{array}{l} t_0 := 1 - \cos x\\ t_1 := \frac{t_0}{\sin x}\\ \mathbf{if}\;t_1 \leq -0.02:\\ \;\;\;\;\frac{\cos x \cdot t_0}{\cos x \cdot \sin x}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;0.5 \cdot x + 0.041666666666666664 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	31.1
Cost	192

\[0.5 \cdot x \]

Reproduce

herbie shell --seed 2023104 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64

  :herbie-target
  (tan (/ x 2.0))

  (/ (- 1.0 (cos x)) (sin x)))