Average Error: 14.7 → 0.1
Time: 9.7s
Precision: binary64
Cost: 6720
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
↓
\[\frac{\tan \left(x \cdot 0.5\right)}{0.75} \]
(FPCore (x)
 :precision binary64
 (/ (* (* (/ 8.0 3.0) (sin (* x 0.5))) (sin (* x 0.5))) (sin x)))
↓
(FPCore (x) :precision binary64 (/ (tan (* x 0.5)) 0.75))
double code(double x) {
	return (((8.0 / 3.0) * sin((x * 0.5))) * sin((x * 0.5))) / sin(x);
}
↓
double code(double x) {
	return tan((x * 0.5)) / 0.75;
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((8.0d0 / 3.0d0) * sin((x * 0.5d0))) * sin((x * 0.5d0))) / sin(x)
end function
↓
real(8) function code(x)
    real(8), intent (in) :: x
    code = tan((x * 0.5d0)) / 0.75d0
end function

public static double code(double x) {
	return (((8.0 / 3.0) * Math.sin((x * 0.5))) * Math.sin((x * 0.5))) / Math.sin(x);
}
↓
public static double code(double x) {
	return Math.tan((x * 0.5)) / 0.75;
}

def code(x):
	return (((8.0 / 3.0) * math.sin((x * 0.5))) * math.sin((x * 0.5))) / math.sin(x)
↓
def code(x):
	return math.tan((x * 0.5)) / 0.75

function code(x)
	return Float64(Float64(Float64(Float64(8.0 / 3.0) * sin(Float64(x * 0.5))) * sin(Float64(x * 0.5))) / sin(x))
end
↓
function code(x)
	return Float64(tan(Float64(x * 0.5)) / 0.75)
end

function tmp = code(x)
	tmp = (((8.0 / 3.0) * sin((x * 0.5))) * sin((x * 0.5))) / sin(x);
end
↓
function tmp = code(x)
	tmp = tan((x * 0.5)) / 0.75;
end

code[x_] := N[(N[(N[(N[(8.0 / 3.0), $MachinePrecision] * N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]
↓
code[x_] := N[(N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / 0.75), $MachinePrecision]

\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}
↓
\frac{\tan \left(x \cdot 0.5\right)}{0.75}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.7
Target0.3
Herbie0.1
\[\frac{\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

Derivation

  1. Initial program 14.7

    \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

  2. Simplified14.7

    \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \cdot 2.6666666666666665} \]
    Proof
    (*.f64 (/.f64 (*.f64 (sin.f64 (*.f64 x 1/2)) (sin.f64 (*.f64 x 1/2))) (sin.f64 x)) 8/3): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (*.f64 (sin.f64 (*.f64 x 1/2)) (sin.f64 (*.f64 x 1/2))) (sin.f64 x)) (Rewrite<= metadata-eval (/.f64 8 3))): 1 points increase in error, 0 points decrease in error
    (Rewrite=> *-commutative_binary64 (*.f64 (/.f64 8 3) (/.f64 (*.f64 (sin.f64 (*.f64 x 1/2)) (sin.f64 (*.f64 x 1/2))) (sin.f64 x)))): 0 points increase in error, 4 points decrease in error
    (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (/.f64 8 3) (*.f64 (sin.f64 (*.f64 x 1/2)) (sin.f64 (*.f64 x 1/2)))) (sin.f64 x))): 5 points increase in error, 0 points decrease in error
    (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 8 3) (sin.f64 (*.f64 x 1/2))) (sin.f64 (*.f64 x 1/2)))) (sin.f64 x)): 0 points increase in error, 5 points decrease in error

  3. Applied egg-rr39.9

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\cos 0 - \cos x}{0.75}}{\sin x}\right)} - 1} \]

  4. Simplified30.6

    \[\leadsto \color{blue}{\frac{1 - \cos x}{\sin x \cdot 0.75}} \]
    Proof
    (/.f64 (-.f64 1 (cos.f64 x)) (*.f64 (sin.f64 x) 3/4)): 0 points increase in error, 0 points decrease in error
    (/.f64 (-.f64 (Rewrite<= cos-0_binary64 (cos.f64 0)) (cos.f64 x)) (*.f64 (sin.f64 x) 3/4)): 3 points increase in error, 2 points decrease in error
    (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (-.f64 (cos.f64 0) (cos.f64 x)) 3/4) (sin.f64 x))): 5 points increase in error, 0 points decrease in error
    (Rewrite<= expm1-log1p_binary64 (expm1.f64 (log1p.f64 (/.f64 (/.f64 (-.f64 (cos.f64 0) (cos.f64 x)) 3/4) (sin.f64 x))))): 0 points increase in error, 5 points decrease in error
    (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (log1p.f64 (/.f64 (/.f64 (-.f64 (cos.f64 0) (cos.f64 x)) 3/4) (sin.f64 x)))) 1)): 5 points increase in error, 0 points decrease in error

  5. Applied egg-rr0.3

    \[\leadsto \color{blue}{{\left(\frac{0.75}{\tan \left(\frac{x}{2}\right)}\right)}^{-1}} \]

  6. Applied egg-rr0.1

    \[\leadsto \color{blue}{\frac{\tan \left(x \cdot 0.5\right)}{0.75}} \]

  7. Final simplification0.1

    \[\leadsto \frac{\tan \left(x \cdot 0.5\right)}{0.75} \]

Alternatives

Alternative 1
Error0.4
Cost6720
\[\tan \left(\frac{x}{2}\right) \cdot 1.3333333333333333 \]
Alternative 2
Error31.0
Cost6656
\[{\left(\frac{1.5}{x}\right)}^{-1} \]
Alternative 3
Error31.0
Cost320
\[\frac{0.2962962962962963}{\frac{0.4444444444444444}{x}} \]
Alternative 4
Error31.1
Cost192
\[x \cdot 0.6666666666666666 \]

Error

Reproduce

herbie shell --seed 2022343 
(FPCore (x)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A"
  :precision binary64

  :herbie-target
  (/ (/ (* 8.0 (sin (* x 0.5))) 3.0) (/ (sin x) (sin (* x 0.5))))

  (/ (* (* (/ 8.0 3.0) (sin (* x 0.5))) (sin (* x 0.5))) (sin x)))